Skip to main content
permalink

Section 11.1 Function Basics

permalink

permalinkIn this section, we will introduce a topic that will be essential for continued mathematical learning: functions. Functions should be thought of as machines that turn one number into another number, much like a cash register can turn a number of pounds of fruit into a price.

permalink
Figure 11.1.1. Alternative Video Lesson
permalink

Subsection 11.1.1 Informal Definition of a Function

We are familiar with the xx symbol. This symbol is used to turn numbers into their square roots. Sometimes it's simple to do this on paper or in our heads, and sometimes it helps to have a calculator. We can see some calculations in Figure 11.1.2.

permalink
99 =3=3
1/41/4 =1/2=1/2
22 1.411.41
Figure 11.1.2. Values of xx

permalinkThe xx symbol represents a process; it's a way for us to turn numbers into other numbers. This idea of having a process for turning numbers into other numbers is the fundamental topic of this chapter.

permalink
Definition 11.1.3. Function (Informal Definition).

A function is a process for turning numbers into (potentially) different numbers. The process must be consistent, in that whenever you apply it to some particular number, you always get the same result.

permalinkSection 11.5 covers a more technical definition for functions, and covers topics that are more appropriate when using that definition. Definition 11.1.3 is so broad that you probably use functions all the time.

permalink
Example 11.1.4.

In each of these examples, some process is used for turning one number into another.

  • If you convert a person's birth year into their age, you are using a function.

  • If you look up the Kelly Blue Book value of a Honda Odyssey based on how old it is, you are using a function.

  • If you use the expected guest count for a party to determine how many pizzas you should order, you are using a function.

The xx function is consistent; for example, every time you evaluate 9,9, you always get 3. One interesting fact is that xx is not found on most keyboards, and yet computers can still find square roots. Computer technicians write sqrt(x)sqrt(x) when they want to compute a square root, as we see in Figure 11.1.5.

permalink
sqrt(9)sqrt(9) =3=3
sqrt(1/4)sqrt(1/4) =1/2=1/2
sqrt(2)sqrt(2) 1.411.41
Figure 11.1.5. Values of sqrt(x)sqrt(x)

permalinkThe parentheses in sqrt(x)sqrt(x) are very important. To see why, try to put yourself in the “mind” of a computer. The computer will recognize sqrt and know that it needs to compute a square root but without parentheses it will think that it needs to compute sqrt4 and then put a 9 on the end, which would produce a final result of 29.29. This is probably not what was intended. And so the purpose of the parentheses in sqrt(49) is to be deliberately clear.

permalinkFunctions have their own names. We've seen a function named sqrt,sqrt, but any name you can imagine is allowable. In the sciences, it is common to name functions with whole words, like weightweight or health_index.health_index. In math, we often abbreviate such function names to ww or h.h. And of course, since the word “function” itself starts with “f,” we will often name a function f.f.

permalink
Warning 11.1.6. Notation Ambiguity.

In some contexts, the symbol tt might represent a variable (a number that is represented by a letter) and in other contexts, tt might represent a function (a process for changing numbers into other numbers). By staying conscious of the context of an investigation, we avoid confusion.

permalinkNext we need to discuss how we go about using a function's name.

permalink
Definition 11.1.7. Function Notation.

The standard notation for referring to functions involves giving the function itself a name, and then writing:

nameoffunction(input)nameoffunctioninput
permalink
Example 11.1.8.

f(13)f(13) is pronounced “f of 13.” The word “of” is very important, because it reminds us that ff is a process and we are about to apply that process to the input value 13.13. So ff is the function, 1313 is the input, and f(13)f(13) is the output we'd get from using 1313 as input.

f(x)f(x) is pronounced “f of x.” This is just like the previous example, except that the input is not any specific number. The value of xx could be 1313 or any other number. Whatever xx's value, f(x)f(x) means the corresponding output from the function f.f.

BudgetDeficit(2017)BudgetDeficit(2017) is pronounced “BudgetDeficit of 2017.” This is probably about a function that takes a year as input, and gives that year's federal budget deficit as output. The process here of changing a year into a dollar amount might not involve any mathematical formula, but rather looking up information from the Congressional Budget Office's website.

permalink
Note 11.1.9.

While a function has a name like f,f, and the input to that function often has a variable name like x,x, the expression f(x)f(x) represents the output of the function. To be clear, f(x)f(x) is not a function. Rather, ff is a function, and f(x)f(x) its output when the number xx was used as input.

permalink
Checkpoint 11.1.10.
permalink
Warning 11.1.11. More Notation Ambiguity.

As mentioned in Warning 11.1.6, we need to remain conscious of the context of any symbol we are using. Consider the expression a(b).a(b). This could easily mean the output of a function aa with input b.b. It could also mean that two numbers aa and bb need to be multiplied. It all depends on the context in which these symbols are being used.

Sometimes it's helpful to think of a function as a machine, as in Figure 11.1.12. A function has the capacity to take in all kinds of different numbers into it's hopper (feeding tray) as inputs and transform them into their outputs.

permalinkA box labeled function machine showing spheres as inputs going into the  hopper; there is an output chute with cubes coming out.
Figure 11.1.12. Imagining a function as a machine. (Image by Duane Nykamp using Mathematica.)
permalink

Subsection 11.1.2 Tables and Graphs

permalinkSince functions are potentially complicated, we want ways to understand them more easily. Two basic tools for understanding a function better are tables and graphs.

permalink
Example 11.1.13. A Table for the Budget Deficit Function.

Consider the function BudgetDeficit,BudgetDeficit, that takes a year as its input and outputs the US federal budget deficit for that year. For example, the Congressional Budget Office's website tells us that BudgetDeficit(2009)BudgetDeficit(2009) is $1.41$1.41 trillion. If we'd like to understand this function better, we might make a table of all the inputs and outputs we can find. Using the CBO's website (www.cbo.gov/topics/budget), we can put together Table 11.1.14.

permalink
input
xx (year)
output
BudgetDeficit(x)BudgetDeficit(x) ($trillion)
2007 0.160.16
2008 0.460.46
2009 1.41.4
2010 1.31.3
2011 1.31.3
2012 1.11.1
2013 0.680.68
2014 0.480.48
2015 0.440.44
2016 0.590.59
2017 0.670.67
2018 0.780.78
Table 11.1.14. The Federal Budget Deficit

How is this table helpful? There are things about the function that we can see now by looking at the numbers in this table.

  • We can see that the budget deficit had a spike between 2008 and 2009.

  • And it fell again between 2012 and 2013.

  • It appears to stay roughly steady for several years at a time, with occasional big jumps or drops.

These observations help us understand the function BudgetDeficitBudgetDeficit a little better.

permalink
Checkpoint 11.1.15.
permalink
Example 11.1.16. A Table for the Square Root Function.

Let's return to our example of the function sqrt.sqrt. Tabulating some inputs and outputs reveals Figure 11.1.17.

permalink
input, xx output, sqrt(x)sqrt(x)
00 00
11 11
22 1.411.41
33 1.731.73
44 22
55 2.242.24
66 2.452.45
77 2.652.65
88 2.832.83
99 33
Figure 11.1.17.

How is this table helpful? Here are some observations that we can make now.

  • We can see that when input numbers increase, so do output numbers.

  • We can see even though outputs are increasing, they increase by less and less with each step forward in x.x.

These observations help us understand sqrtsqrt a little better. For instance, based on these observations which do you think is larger: the difference between sqrt(23)sqrt(23) and sqrt(24),sqrt(24), or the difference between sqrt(85)sqrt(85) and sqrt(86)?sqrt(86)?

permalink
Checkpoint 11.1.18.

permalinkAnother powerful tool for understanding a function better is a graph. Given a function f,f, one way to make its graph is to take a table of input and output values, and read each row as the coordinates of a point in the xyxy-plane.

permalink
Example 11.1.19. A Graph for the Budget Deficit Function.

Returning to the function BudgetDeficitBudgetDeficit that we studied in Example 11.1.13, in order to make a graph of this function we view Table 11.1.14 as a list of points with xx and yy coordinates, as in Figure 11.1.20. We then plot these points on a set of coordinate axes, as in Figure 11.1.21. The points have been connected with a curve so that we can see the overall pattern given by the progression of points. Since there was not any actual data for inputs in between any two years, the curve is dashed. That is, this curve is dashed because it just represents someone's best guess as to how to connect the plotted points. Only the plotted points themselves are precise.

permalink
(input,output)(input,output)
(x,BudgetDeficit(x))(x,BudgetDeficit(x))
(2007,0.16)(2007,0.16)
(2008,0.46)(2008,0.46)
(2009,1.4)(2009,1.4)
(2010,1.3)(2010,1.3)
(2011,1.3)(2011,1.3)
(2012,1.1)(2012,1.1)
(2013,0.68)(2013,0.68)
(2014,0.48)(2014,0.48)
(2015,0.44)(2015,0.44)
(2016,0.59)(2016,0.59)
(2017,0.67)(2017,0.67)
(2018,0.78)(2018,0.78)
Figure 11.1.20.
permalinkA plot of the data in the table; the year is on the horizontal axis and the amount of the deficit is on the vertical axis; the points are connected with a dotted line
Figure 11.1.21. y=BudgetDeficit(x)y=BudgetDeficit(x)

How has this graph helped us to understand the function better? All of the observations that we made in Example 11.1.13 are perhaps even more clear now. For instance, the spike in the deficit between 2008 and 2009 is now visually apparent. Seeking an explanation for this spike, we recall that there was a financial crisis in late 2008. Revenue from income taxes dropped at the same time that federal money was spent to prevent further losses.

permalink
Example 11.1.22. A Graph for the Square Root Function.

Let's now construct a graph for sqrt.sqrt. Tabulating inputs and outputs gives the points in Figure 11.1.23, which in turn gives us the graph in Figure 11.1.24.

permalink
(input,output)(input,output)
(x,sqrt(x))(x,sqrt(x))
(0,0)(0,0)
(1,1)(1,1)
(2,1.41)(2,1.41)
(3,1.73)(3,1.73)
(4,2)(4,2)
(5,2.24)(5,2.24)
(6,2.45)(6,2.45)
(7,2.65)(7,2.65)
(8,2.83)(8,2.83)
(9,3)(9,3)
Figure 11.1.23.
permalinka graph of the points from the table connected with a dotted line
Figure 11.1.24. y=sqrt(x)y=sqrt(x)

Just as in the previous example, we've plotted points where we have concrete coordinates, and then we have made our best attempt to connect those points with a curve. Unlike the previous example, here we believe that points will continue to follow the same pattern indefinitely to the right, and so we have added an arrowhead to the graph.

What has this graph done to improve our understanding of sqrt?sqrt? As inputs (xx-values) increase, the outputs (yy-values) increase too, although not at the same rate. In fact we can see that our graph is steep on its left, and less steep as we move to the right. This confirms our earlier observation in Example 11.1.16 that outputs increase by smaller and smaller amounts as the input increases.

permalink
Remark 11.1.25. Graph of a Function.

Given a function f,f, when we refer to a graph of ff we are not referring to an entire picture, like Figure 11.1.24. A graph of ff is only part of that picture—the curve and the points that it connects. Everything else (axes, tick marks, the grid, labels, and the surrounding white space) is just useful decoration so that we can read the graph more easily.

permalink
Remark 11.1.26. A Common Wording Misunderstanding.

It is common to refer to the graph of ff as the graph of the equation y=f(x)y=f(x). However, we should avoid saying “the graph of f(x).f(x).” That would indicate a misunderstanding of our notation. Since f(x)f(x) is the output for a certain input x.x. That means that f(x)f(x) is just a number and not worthy of a two-dimensional picture.

permalinkWhile it is important to be able to make a graph of a function f,f, we also need to be capable of looking at a graph and reading it well. A graph of ff provides us with helpful specific information about f;f; it tells us what ff does to its input values. When we were making graphs, we plotted points of the form

(input,output)(input,output)

permalinkNow given a graph of f,f, we interpret coordinates in the same way.

permalink
Example 11.1.27.

In Figure 11.1.28 we have a graph of a function f.f. If we wish to find f(1),f(1), we recognize that 11 is being used as an input. So we would want to find a point of the form (1,y).(1,y). Seeking out xx-coordinate 11 in Figure 11.1.28, we find that the only such point is (1,2).(1,2). Therefore the output for 11 is 2;2; in other words f(1)=2.f(1)=2.

permalink
Figure 11.1.28. y=f(x)y=f(x)
permalink
Checkpoint 11.1.29.
permalink
Example 11.1.30. Unemployment Rates.

Suppose that uu is the unemployment function of time. That is, u(t)u(t) is the unemployment rate in the United States in year t.t. The graph of the equation y=u(t)y=u(t) is given in Figure 11.1.31 (data.bls.gov/timeseries/LNS14000000).

permalinkA time series plot of the unemployment data from 2007 to 2017; the points on the graph are (2007, 4.62), (2008, 5.80), (2009, 9.28), (2010, 9.61), (2011, 8.93), (2012,8.08), (2013,7.36), (2014,6.18), (2015,5.27), (2016,4.87), (2017,4.35) and (2018,3.89); the points are connected with a dotted line
Figure 11.1.31. Unemployment in the United States

What was the unemployment in 2008? It is a straightforward matter to use Figure 11.1.31 to find that unemployment was almost 6%6% in 2008. Asking this question is exactly the same thing as asking to find u(2008).u(2008). That is, we have one question that can either be asked in an everyday-English way or which can be asked in a terse, mathematical notation-heavy way:

“What was unemployment in 2008?”

“Find u(2008).u(2008).

If we use the table to establish that u(2009)9.25,u(2009)9.25, then we should be prepared to translate that into everyday-English using the context of the function: In 2009, unemployment in the U.S. was about 9.25%.9.25%.

If we ask the question “when was unemployment at 5%,5%,” we can read the graph and see that there were two such times: mid-2007 and about 2016. But there is again a more mathematical notation-heavy way to ask this question. Namely, since we are being told that the output of uu is 5,5, we are being asked to solve the equation u(t)=5.u(t)=5. So the following communicate the same thing:

“When was unemployment at 5%?5%?

“Solve the equation u(t)=5.u(t)=5.

And our answer to this question is:

“Unemployment was at 5%5% in mid-2007 and about 2016.”

t2007.5t2007.5 or t2016.t2016.

permalink
Checkpoint 11.1.32.
permalink

Subsection 11.1.3 Translating Between Four Descriptions of the Same Function

permalinkWe have noted that functions are complicated, and we want ways to make them easier to understand. It's common to find a problem involving a function and not know how to find a solution to that problem. Most functions have at least four standard ways to think about them, and if we learn how to translate between these four perspectives, we often find that one of them makes a given problem easier to solve.

The four modes for working with a given function are

  • a verbal description

  • a table of inputs and outputs

  • a graph of the function

  • a formula for the function

This has been visualized in Figure 11.1.33.

permalinka diagram with "a function, f," written in the center; around the cirle there is a rectangle for each mode of working with a function: verbal, table, graph and formula; the rectangles for each mode are all connected by a larger circle with arrows pointing in each direction
Figure 11.1.33. Function Perspectives
permalink
Example 11.1.34.

Consider a function ff that squares its input and then adds 1.1. Translate this verbal description of ff into a table, a graph, and a formula.

Explanation

To make a table for \(f\text{,}\) we'll have to select some input \(x\)-values. These choices are left entirely up to us, so we might as well choose small, easy-to-work-with values. However we shouldn't shy away from negative input values. Given the verbal description, we should be able to compute a column of output values. Figure 11.1.35 is one possible table that we might end up with.

permalink
\(x\) \(f(x)\)
\(-2\) \((-2)^2+1=5\)
\(-1\) \((-1)^2+1=2\)
\(0 \) \(0^2+1=1\)
\(1 \) \(1^2+1=2\)
\(2 \) \(5\)
\(3 \) \(10\)
\(4 \) \(17\)
Figure 11.1.35.

Once we have a table for \(f\text{,}\) we can make a graph for \(f\) as in Figure 11.1.36, using the table to plot points.

permalinka graph of the points listed in the table; the points are connected with a dotted line and form a parabola
Figure 11.1.36. \(y=f(x)\)

Lastly, we must find a formula for \(f\text{.}\) This means we need to write an algebraic expression that says the same thing about \(f\) as the verbal description, the table, and the graph. For this example, we can focus on the verbal description. Since \(f\) takes its input, squares it, and adds \(1\text{,}\) we have that

\begin{equation*} f(x)=x^2+1\text{.} \end{equation*}
permalink
Example 11.1.37.

Let FF be the function that takes a Celsius temperature as input and outputs the corresponding Fahrenheit temperature. Translate this verbal description of FF into a table, a graph, and a formula.

Explanation

To make a table for \(F\text{,}\) we will need to rely on what we know about Celsius and Fahrenheit temperatures. It is a fact that the freezing temperature of water at sea level is 0 °C, which equals 32 °F. Also, the boiling temperature of water at sea level is 100 °C, which is the same as 212 °F. One more piece of information we might have is that standard human body temperature is 37 °C, or 98.6 °F. All of this is compiled in Figure 11.1.38. Note that we tabulated inputs and outputs by working with the context of the function, not with any computations.

permalink
\(C\) \(F(C)\)
\(0 \) \(32\)
\(37 \) \(98.6\)
\(100 \) \(212\)
Figure 11.1.38.

Once a table is established, making a graph by plotting points is a simple matter, as in Figure 11.1.39. The three plotted points seem to be in a straight line, so we think it is reasonable to connect them in that way.

permalinka plot of the points listed in the table; the points are connected by a straight, dotted line
Figure 11.1.39. \(y=F(C)\)

To find a formula for \(F\text{,}\) the verbal definition is not of much direct help. But \(F\)'s graph does seem to be a straight line. And linear equations are familiar to us. This line has a \(y\)-intercept at \((0,32)\) and a slope we can calculate: \(\frac{212-32}{100-0}=\frac{180}{100}=\frac{9}{5}\text{.}\) So the equation of this line is \(y=\frac{9}{5}C+32\text{.}\) On the other hand, the equation of this graph is \(y=F(C)\text{,}\) since it is a graph of the function \(F\text{.}\) So evidently,

\begin{equation*} F(C)=\frac{9}{5}C+32\text{.} \end{equation*}
permalink

Reading Questions 11.1.4 Reading Questions

permalink
1.

When gg is a function, how should you say out loud “g(x)?g(x)?

permalink
2.

There are four main ways to communicate how a function turns its inputs into its outputs. What are they?

permalink
3.

What is usually an acceptable way to type “the square root of xx” if you have to type it using a regular keyboard?

permalink

Exercises 11.1.5 Exercises

Review and Warmup
permalink
1.

Locate each point in the graph:

Write each point’s position as an ordered pair, like (1,2).(1,2).

A=A= B=B=
C=C= D=D=
permalink
2.

Locate each point in the graph:

Write each point’s position as an ordered pair, like (1,2).(1,2).

A=A= B=B=
C=C= D=D=
permalink
3.

Evaluate 2t18t2t18t for t=4.t=4.

permalink
4.

Evaluate 6t48t6t48t for t=10.t=10.

permalink
5.
  1. Evaluate 2x22x2 when x=4.x=4.

  2. Evaluate (2x)2(2x)2 when x=4.x=4.

permalink
6.
  1. Evaluate 3x23x2 when x=2.x=2.

  2. Evaluate (3x)2(3x)2 when x=2.x=2.

Function Formulas and Evaluation
permalink
7.

Evaluate the function at the given values.

H(x)=x4H(x)=x4

  1. H(5)=H(5)=

  2. H(2)=H(2)=

  3. H(0)=H(0)=

permalink
8.

Evaluate the function at the given values.

G(x)=x1G(x)=x1

  1. G(3)=G(3)=

  2. G(5)=G(5)=

  3. G(0)=G(0)=

permalink
9.

Evaluate the function at the given values.

F(x)=4xF(x)=4x

  1. F(3)=F(3)=

  2. F(4)=F(4)=

  3. F(0)=F(0)=

permalink
10.

Evaluate the function at the given values.

G(x)=10xG(x)=10x

  1. G(1)=G(1)=

  2. G(5)=G(5)=

  3. G(0)=G(0)=

permalink
11.

Evaluate the function at the given values.

H(x)=3x+5H(x)=3x+5

  1. H(4)=H(4)=

  2. H(2)=H(2)=

  3. H(0)=H(0)=

permalink
12.

Evaluate the function at the given values.

K(x)=5x+9K(x)=5x+9

  1. K(3)=K(3)=

  2. K(2)=K(2)=

  3. K(0)=K(0)=

permalink
13.

Evaluate the function at the given values.

K(x)=x+9K(x)=x+9

  1. K(2)=K(2)=

  2. K(5)=K(5)=

  3. K(0)=K(0)=

permalink
14.

Evaluate the function at the given values.

f(x)=x+6f(x)=x+6

  1. f(4)=f(4)=

  2. f(1)=f(1)=

  3. f(0)=f(0)=

permalink
15.

Evaluate the function at the given values.

g(x)=x28g(x)=x28

  1. g(4)=g(4)=

  2. g(2)=g(2)=

  3. g(0)=g(0)=

permalink
16.

Evaluate the function at the given values.

h(t)=t23h(t)=t23

  1. h(4)=h(4)=

  2. h(4)=h(4)=

  3. h(0)=h(0)=

permalink
17.

Evaluate the function at the given values.

F(y)=y2+9F(y)=y2+9

  1. F(1)=F(1)=

  2. F(2)=F(2)=

  3. F(0)=F(0)=

permalink
18.

Evaluate the function at the given values.

F(x)=x23F(x)=x23

  1. F(5)=F(5)=

  2. F(2)=F(2)=

  3. F(0)=F(0)=

permalink
19.

Evaluate the function at the given values.

G(t)=6G(t)=6

  1. G(3)=G(3)=

  2. G(6)=G(6)=

  3. G(0)=G(0)=

permalink
20.

Evaluate the function at the given values.

H(y)=7H(y)=7

  1. H(2)=H(2)=

  2. H(7)=H(7)=

  3. H(0)=H(0)=

permalink
21.

Evaluate the function at the given values.

K(x)=7x2x+10K(x)=7x2x+10

  1. K(5)=K(5)= .

  2. K(1)=K(1)= .

permalink
22.

Evaluate the function at the given values.

K(x)=7x10x+4K(x)=7x10x+4

  1. K(5)=K(5)= .

  2. K(6)=K(6)= .

permalink
23.

Evaluate the function at the given values.

f(x)=3x2f(x)=3x2 .

  1. f(1)=f(1)= .

  2. f(2)=f(2)= .

permalink
24.

Evaluate the function at the given values.

g(x)=70x7g(x)=70x7 .

  1. g(14)=g(14)= .

  2. g(7)=g(7)= .

permalink
25.

Evaluate the function at the given values.

h(x)=3x4h(x)=3x4

  1. h(7)=h(7)=

  2. h(4)=h(4)=

permalink
26.

Evaluate the function at the given values.

F(x)=6x+3F(x)=6x+3

  1. F(5)=F(5)=

  2. F(5)=F(5)=

permalink
27.

Evaluate the function at the given values.

F(x)=x2+5x5F(x)=x2+5x5

  1. F(1)=F(1)=

  2. F(3)=F(3)=

permalink
28.

Evaluate the function at the given values.

G(x)=x2+2xG(x)=x2+2x

  1. G(0)=G(0)=

  2. G(3)=G(3)=

permalink
29.

Evaluate the function at the given values.

H(x)=3x2+5x+3H(x)=3x2+5x+3

  1. H(2)=H(2)=

  2. H(2)=H(2)=

permalink
30.

Evaluate the function at the given values.

K(x)=2x22x1K(x)=2x22x1

  1. K(1)=K(1)=

  2. K(3)=K(3)=

permalink
31.

Evaluate the function at the given values.

K(x)=x.K(x)=x.

  1. K(49)=K(49)=

  2. K(649)=K(649)=

  3. K(6)=K(6)=

permalink
32.

Evaluate the function at the given values.

f(x)=x.f(x)=x.

  1. f(16)=f(16)=

  2. f(49)=f(49)=

  3. f(6)=f(6)=

permalink
33.

Evaluate the function at the given values.

g(x)=3xg(x)=3x

  1. g(1)=g(1)=

  2. g(6427)=g(6427)=

permalink
34.

Evaluate the function at the given values.

h(x)=3xh(x)=3x

  1. h(27)=h(27)=

  2. h(127)=h(127)=

permalink
35.

Evaluate the function at the given values.

F(x)=12F(x)=12

  1. F(4)=

  2. F(8)=

permalink
36.

Evaluate the function at the given values.

F(x)=15

  1. F(7)=

  2. F(2)=

Function Formulas and Solving Equations
permalink
37.

Solve for x, where G(x)=12x+6.

  1. G(x)=42

  2. G(x)=3

permalink
38.

Solve for x, where H(x)=8x5.

  1. H(x)=19

  2. H(x)=7

permalink
39.

Solve for x, where K(x)=x2+7.

  1. K(x)=8

  2. K(x)=6

permalink
40.

Solve for x, where K(x)=x21.

  1. K(x)=8

  2. K(x)=4

permalink
41.

Solve for x, where f(x)=x2+x73.

f(x)=1

permalink
42.

Solve for x, where g(x)=x2+3x25.

g(x)=7

permalink
43.

If h is a function defined by h(y)=4y+9,

  1. Find h(0).

  2. Solve h(y)=0.

permalink
44.

If f is a function defined by f(y)=4y+2,

  1. Find f(0).

  2. Solve f(y)=0.

permalink
45.

If H is a function defined by H(r)=4r24,

  1. Find H(0).

  2. Solve H(r)=0.

permalink
46.

If h is a function defined by h(r)=r21,

  1. Find h(0).

  2. Solve h(r)=0.

permalink
47.

If f is a function defined by f(t)=t29t+18,

  1. Find f(0).

  2. Solve f(t)=0.

permalink
48.

If G is a function defined by G(t)=t2+2t35,

  1. Find G(0).

  2. Solve G(t)=0.

Functions and Points on a Graph
permalink
49.
  1. If K(4)=2, then the point is on the graph of K.

  2. If (3,0) is on the graph of K, then K(3)=.

permalink
50.
  1. If f(10)=12, then the point is on the graph of f.

  2. If (10,7) is on the graph of f, then f(10)=.

permalink
51.

If g(r)=x, then the point is on the graph of g.

permalink
52.

If h(y)=r, then the point is on the graph of h.

permalink
53.

If (t,x) is on the graph of h, then h(t)=.

permalink
54.

If (r,y) is on the graph of F, then F(r)=.

permalink
55.

For the function G, when x=1, the output is 0.

Choose all true statements.

  • G(1)=0

  • G(0)=1

  • The function's value is 1 at 0.

  • The point (1,0) is on the graph of the function.

  • The function's value is 0 at 1.

  • The point (0,1) is on the graph of the function.

permalink
56.

For the function H, when x=2, the output is 11.

Choose all true statements.

  • The function's value is 2 at 11.

  • The point (11,2) is on the graph of the function.

  • H(11)=2

  • The point (2,11) is on the graph of the function.

  • The function's value is 11 at 2.

  • H(2)=11

Function Graphs
permalink
57.

Use the graph of K below to evaluate the given expressions. (Estimates are OK.)

  1. K(1)=

  2. K(3)=

permalink
58.

Use the graph of K below to evaluate the given expressions. (Estimates are OK.)

  1. K(3)=

  2. K(1)=

permalink
59.

Use the graph of f below to evaluate the given expressions. (Estimates are OK.)

  1. f(2)=

  2. f(7)=

permalink
60.

Use the graph of g below to evaluate the given expressions. (Estimates are OK.)

  1. g(4)=

  2. g(0)=

permalink
61.

Use the graph of h below to evaluate the given expressions. (Estimates are OK.)

  1. h(2)=

  2. h(1)=

permalink
62.

Use the graph of h below to evaluate the given expressions. (Estimates are OK.)

  1. h(2)=

  2. h(3)=

permalink
63.

Function f is graphed.

  1. Find f(5).

  2. Solve f(x)=0.

permalink
64.

Function f is graphed.

  1. Find f(2).

  2. Solve f(x)=2.

permalink
65.

Function f is graphed.

  1. Find f(2)=.

  2. Solve f(x)=3.

permalink
66.

Function f is graphed.

  1. Find f(3)=.

  2. Solve f(x)=0.

permalink
67.

Function f is graphed.

  1. Find f(1)=.

  2. Solve f(x)=1.

permalink
68.

Function f is graphed.

  1. Find f(1)=.

  2. Solve f(x)=2.

Function Tables
permalink
69.

Use the table of values for g below to evaluate the given expressions.

x 0 2 4 6 8
g(x) 6.1 9.3 6.2 9.7 4.5
  1. g(2)=

  2. g(8)=

permalink
70.

Use the table of values for h below to evaluate the given expressions.

x 1 0 1 2 3
h(x) 2.8 9.1 0.1 1.4 9.4
  1. h(0)=

  2. h(2)=

permalink
71.

Make a table of values for the function G, defined by G(x)=2x2. Based on values in the table, sketch a graph of G.

x G(x)
permalink
72.

Make a table of values for the function H, defined by H(x)=2x+2x2+3. Based on values in the table, sketch a graph of H.

x H(x)
Translating Between Different Representations of a Function
permalink
73.

Here is a verbal description of a function G.

“Cube the input x to obtain the output y.

  1. Give a numeric representation of G.

    x 0 1 2 3 4
    G(x)
  2. Give a formula for G.

permalink
74.

Here is a verbal description of a function H.

“Cube the input x to obtain the output y.

  1. Give a numeric representation of H.

    x 0 1 2 3 4
    H(x)
  2. Give a formula for H.

permalink
75.

Here is a verbal description of a function K.

“Double the input x and then subtract three to obtain the output y.

  1. Give a numeric representation of K:

    x 0 1 2 3 4
    K(x)
  2. Give a formula for K.

permalink
76.

Here is a verbal description of a function K.

“Quadruple the input x and then subtract seven to obtain the output y.

  1. Give a numeric representation of K:

    x 0 1 2 3 4
    K(x)
  2. Give a formula for K.

permalink
77.

Express the function f numerically with the table.

f(x)=2x312x2
x 3 2 1 0 1 2 3
f(x)

On graphing paper, you should be able to give a graphical representation of f too.

permalink
78.

Express the function g numerically with the table.

g(x)=x212x
x 3 2 1 0 1 2 3
g(x)

On graphing paper, you should be able to give a graphical representation of g too.

permalink
79.

Express the function h numerically with the table.

h(x)=8x7+x
x 3 2 1 0 1 2 3
h(x)

On graphing paper, you should be able to give a graphical representation of h too.

permalink
80.

Express the function h numerically with the table.

h(x)=5x4+x
x 3 2 1 0 1 2 3
h(x)

On graphing paper, you should be able to give a graphical representation of h too.

Functions in Context
permalink
81.

Phil started saving in a piggy bank on his birthday. The function f(x)=2x+2 models the amount of money, in dollars, in Phil’s piggy bank. The independent variable represents the number of days passed since his birthday.

Interpret the meaning of f(4)=10.

  • A. Four days after Phil started his piggy bank, there were $10 in it.

  • B. The piggy bank started with $10 in it, and Phil saves $4 each day.

  • C. The piggy bank started with $4 in it, and Phil saves $10 each day.

  • D. Ten days after Phil started his piggy bank, there were $4 in it.

permalink
82.

An arcade sells multi-day passes. The function g(x)=13x models the number of days a pass will work, where x is the amount of money paid, in dollars.

Interpret the meaning of g(12)=4.

  • A. Each pass costs $12, and it works for 4 days.

  • B. If a pass costs $4, it will work for 12 days.

  • C. If a pass costs $12, it will work for 4 days.

  • D. Each pass costs $4, and it works for 12 days.

permalink
83.

Maygen will spend $175 to purchase some bowls and some plates. Each bowl costs $3, and each plate costs $5. The function p(b)=35b+35 models the number of plates Maygen will purchase, where b represents the number of bowls Maygen will purchase.

Interpret the meaning of p(45)=8.

  • A. If 45 bowls are purchased, then 8 plates will be purchased.

  • B. $8 will be used to purchase bowls, and $45 will be used to purchase plates.

  • C. If 8 bowls are purchased, then 45 plates will be purchased.

  • D. $45 will be used to purchase bowls, and $8 will be used to purchase plates.

permalink
84.

Carly will spend $450 to purchase some bowls and some plates. Each plate costs $2, and each bowl costs $9. The function q(x)=29x+50 models the number of bowls Carly will purchase, where x represents the number of plates to be purchased.

Interpret the meaning of q(27)=44.

  • A. 44 plates and 27 bowls can be purchased.

  • B. $44 will be used to purchase bowls, and $27 will be used to purchase plates.

  • C. $27 will be used to purchase bowls, and $44 will be used to purchase plates.

  • D. 27 plates and 44 bowls can be purchased.

permalink
85.

Find a formula for the function f that gives the number of hours in x years.

permalink
86.

Find a formula for the function f that gives the number of minutes in x days.

permalink
87.

Suppose that M is the function that computes how many miles are in x feet. Find the formula for M. If you do not know how many feet are in one mile, you can look it up on Google.

Evaluate M(13000) and interpret the result.

There are about miles in feet.

permalink
88.

Suppose that K is the function that computes how many kilograms are in x pounds. Find the formula for K. If you do not know how many pounds are in one kilogram, you can look it up on Google.

Evaluate K(159) and interpret the result.

Something that weighs pounds would weigh about kilograms.

permalink
89.

Suppose that f is the function that the phone company uses to determine what your bill will be (in dollars) for a long-distance phone call that lasts t minutes. Each call costs a fixed price of $2.65 plus 11 cents per minute. Write a formula for this linear function f.

permalink
90.

Suppose that f is the function that gives the total cost (in dollars) of downhill skiing x times during a season with a $500 season pass. Write a formula for f.

permalink
91.

Suppose that f is the function that tells you how many dimes are in x dollars. Write a formula for f.

permalink
92.

The function C models the the number of customers in a store t hours since the store opened.

t 0 1 2 3 4 5 6 7
C(t) 0 40 78 95 99 78 39 0
  1. Find C(6).

  2. Interpret the meaning of C(6).

    • A. There were 39 customers in the store 6 hours after the store opened.

    • B. In 6 hours since the store opened, the store had an average of 39 customers per hour.

    • C. There were 6 customers in the store 39 hours after the store opened.

    • D. In 6 hours since the store opened, there were a total of 39 customers.

  3. Solve C(t)=78 for t. t=

  4. Interpret the meaning of Part c’s solution(s).

    • A. There were 78 customers in the store 2 hours after the store opened.

    • B. There were 78 customers in the store 2 hours after the store opened, and again 5 hours after the store opened.

    • C. There were 78 customers in the store 5 hours after the store opened.

    • D. There were 78 customers in the store either 2 hours after the store opened, or 5 hours after the store opened.

permalink
93.

Let s(t)=13t23t+200, where s is the position (in mi) of a car driving on a straight road at time t (in hr). The car’s velocity (in mi/hr) at time t is given by v(t)=26t3.

  1. Using function notation, express the car’s position after 1.5 hours. The answer here is not a formula, it’s just something using function notation like f(8).

  2. Where is the car then? The answer here is a number with units.

  3. Use function notation to express the question, “When is the car going 59 (mi(hr?” The answer is an equation that uses function notation; something like f(x)=23. You are not being asked to actually solve the equation, just to write down the equation.

  4. Where is the car when it is going 75 (mi(hr? The answer here is a number with units. You are being asked a question about its position, but have been given information about its speed.

permalink
94.

Let s(t)=13t2+t+100, where s is the position (in mi) of a car driving on a straight road at time t (in hr). The car’s velocity (in mi/hr) at time t is given by v(t)=26t+1.

  1. Using function notation, express the car’s position after 3.4 hours. The answer here is not a formula, it’s just something using function notation like f(8).

  2. Where is the car then? The answer here is a number with units.

  3. Use function notation to express the question, “When is the car going 58 (mi(hr?” The answer is an equation that uses function notation; something like f(x)=23. You are not being asked to actually solve the equation, just to write down the equation.

  4. Where is the car when it is going 27 (mi(hr? The answer here is a number with units. You are being asked a question about its position, but have been given information about its speed.

permalink
95.

Describe your own example of a function that has real context to it. You will need some kind of input variable, like “number of years since 2000” or “weight of the passengers in my car.” You will need a process for using that number to bring about a different kind of number. The process does not need to involve a formula; a verbal description would be great, as would a formula.

Give your function a name. Write the symbol(s) that you would use to represent input. Write the symbol(s) that you would use to represent output.

permalink
96.

The following figure has the graph y=d(t), which models a particle’s distance from the starting line in feet, where t stands for time in seconds since timing started.

  1. Find d(7).

  2. Interpret the meaning of d(7).

    • A. In the first 7 seconds, the particle moved a total of 9 feet.

    • B. The particle was 9 feet away from the starting line 7 seconds since timing started.

    • C. In the first 9 seconds, the particle moved a total of 7 feet.

    • D. The particle was 7 feet away from the starting line 9 seconds since timing started.

  3. Solve d(t)=6 for t. t=

  4. Interpret the meaning of part c’s solution(s).

    • A. The particle was 6 feet from the starting line 2 seconds since timing started, or 8 seconds since timing started.

    • B. The particle was 6 feet from the starting line 2 seconds since timing started, and again 8 seconds since timing started.

    • C. The particle was 6 feet from the starting line 2 seconds since timing started.

    • D. The particle was 6 feet from the starting line 8 seconds since timing started.

permalink
97.

The following figure has the graph y=d(t), which models a particle’s distance from the starting line in feet, where t stands for time in seconds since timing started.

  1. Find d(9).

  2. Interpret the meaning of d(9).

    • A. The particle was 2.5 feet away from the starting line 9 seconds since timing started.

    • B. The particle was 9 feet away from the starting line 2.5 seconds since timing started.

    • C. In the first 9 seconds, the particle moved a total of 2.5 feet.

    • D. In the first 2.5 seconds, the particle moved a total of 9 feet.

  3. Solve d(t)=5 for t. t=

  4. Interpret the meaning of part c’s solution(s).

    • A. The particle was 5 feet from the starting line 1 seconds since timing started.

    • B. The particle was 5 feet from the starting line 1 seconds since timing started, and again 8 seconds since timing started.

    • C. The particle was 5 feet from the starting line 1 seconds since timing started, or 8 seconds since timing started.

    • D. The particle was 5 feet from the starting line 8 seconds since timing started.

permalink
98.

Use the graph of h in the figure to fill in the table.

x 2 1 0 1 2
h(x)
  1. Evaluate h(3)h(0).

  2. Evaluate h(2)h(1).

  3. Evaluate 2h(1).

  4. Evaluate h(0)+3.

permalink
99.

Use the given graph of a function f, along with a,b,c,d,e, and h to answer the following questions. Some answers are points, and should be entered as ordered pairs. Some answers ask you to solve for x, so the answer should be in the form x=...

  1. What are the coordinates of the point P?

  2. What are the coordinates of the point Q?

  3. Evaluate f(b). (The answer is symbolic, not a specific number.)

  4. Solve f(x)=e for x. (The answer is symbolic, not a specific number.)

  5. Suppose c=f(z). Solve the equation z=f(x) for x.