Section 11.6 Functions Chapter Review
¶Subsection 11.6.1 Function Basics
In Section 11.1 we defined functions informally, as well as function notation. We saw functions in four forms: verbal descriptions, formulas, graphs and tables.
Example 11.6.1. Informal Definition of a Function.
Determine whether each example below describes a function.
The area of a circle given its radius.
The number you square to get \(9\text{.}\)
The area of a circle given its radius is a function because there is a set of steps or a formula that changes the radius into the area of the circle. We could write \(A(r)=\pi r^2\text{.}\)
The number you square to get \(9\) is not a function because the process we would apply to get the result does not give a single answer. There are two different answers, \(-3\) and \(3\text{.}\) A function must give a single output for a given input.
Example 11.6.2. Tables and Graphs.
Make a table and a graph of the function \(f\text{,}\) where \(f(x)=x^2\text{.}\)
First we will set up a table with negative and positive inputs and calculate the function values. The values are shown in Figure 11.6.3, which in turn gives us the graph in Figure 11.6.4.
| input, \(x\) | output, \(\operatorname{f}(x)\) |
| \(-3\) | \(9\) |
| \(-2\) | \(4\) |
| \(-1\) | \(1\) |
| \(0\) | \(0\) |
| \(1\) | \(2\) |
| \(2\) | \(4\) |
| \(3\) | \(9\) |
Example 11.6.5. Translating between Four Descriptions of the Same Function.
Consider a function \(f\) that triples its input and then adds \(4\text{.}\) Translate this verbal description of \(f\) into a table, a graph, and a formula.
To make a table for \(f\text{,}\) we'll have to select some input \(x\)-values so we will choose some small negative and positive values that are easy to work with. Given the verbal description, we should be able to compute a column of output values. Table 11.6.6 is one possible table that we might end up with.
| \(x\) | \(f(x)\) |
| \(-2\) | \(3(-2)+4=-2\) |
| \(-1\) | \(3(-1)+4=1\) |
| \(0 \) | \(3(0)+4=4\) |
| \(1 \) | \(3(1)+4=7\) |
| \(2 \) | \(3(2)+4=10\) |
Once we have a table for \(f\text{,}\) we can make a graph for \(f\) as in Figure 11.6.7, using the table to plot points.
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, triples it, and adds \(4\text{,}\) we have the formula
Subsection 11.6.2 Domain and Range
In Section 11.2 we saw the definition of domain and range, and three types of domain restrictions. We also learned how to write the domain and range in interval and set-builder notation.
Example 11.6.8.
Determine the domain of \(p\text{,}\) where \(p(x)=\dfrac{x}{2x-1}\text{.}\)
This is an example of the first type of domain restriction, when you have a variable in the denominator. The denominator cannot equal \(0\) so a bad value for \(x\) would be when
The domain is all real numbers except \(\frac{1}{2}\text{.}\)
Example 11.6.9.
What is the domain of the function \(C\text{,}\) where \(C(x)=\sqrt{2x-3}-5\text{?}\)
This is an example of the second type of domain restriction where the value inside the radical cannot be negative. So the good values for \(x\) would be when
So on a number line, if we wanted to picture the domain of \(C\text{,}\) we would make a sketch like:
The domain is the interval \(\left[\frac{3}{2},\infty\right)\text{.}\)
Example 11.6.10. Range.
The range is the collection of possible numbers that \(q\) can give for output. Figure 11.6.11 displays a graph of \(q\text{,}\) with the range shown as an interval on the \(y\)-axis.
The output values are the \(y\)-coordinates so we can see that the \(y\)-values start from \(1\) and continue downward forever. Therefore the range is \((-\infty,1]\text{.}\)
Subsection 11.6.3 Using Technology to Explore Functions
In Section 11.3 we covered how to find a good graphing window and use it to identify all of the key features of a function. We also learned how to solve equations and inequalities using a graph. Here are some examples for review.
Example 11.6.12. Finding an Appropriate Window.
Graph the function \(t\text{,}\) where \(t(x)=(x+10)^2-15\text{,}\) using technology and find a good viewing window.
After some trial and error we found this window that goes from \(-20\) to \(2\) on the \(x\)-axis and \(-20\) to \(100\) on the \(y\)-axis.
Now we can see the vertex and all of the intercepts and we will identify them in the next example.
Example 11.6.15. Using Technology to Determine Key Features of a Graph.
Use the previous graph in figure 11.6.14 to identify the intercepts, minimum or maximum function value, and the domain and range of the function \(t\text{,}\) where \(t(x)=(x+10)^2-15\text{.}\)
Example 11.6.17. Solving Equations and Inequalities Graphically Using Technology.
Use graphing technology to solve the equation \(t(x)=40\text{,}\) where \(t(x)=(x+10)^2-15\text{.}\)
To solve the equation \(t(x)=40\text{,}\) we need to graph \(y=t(x)\) and \(y=40\) on the same axes and find the \(x\)-values where they intersect.
From the graph we can see that the intersection points are approximately \((-17.4,40)\) and \((-2.58,40)\text{.}\) The solution set is \(\{-17.4,-2.58\}\text{.}\)
Subsection 11.6.4 Simplifying Expressions with Function Notation
In Section 11.4 we learned about the difference between \(f(-x)\) and \(-f(x)\) and how to simplify them. We also learned how to simplify other changes to the input and output like \(f(3x)\) and \(\frac{1}{3}f(x)\text{.}\) Here are some examples.
Example 11.6.19. Negative Signs in and out of Function Notation.
Find and simplify a formula for \(f(-x)\) and \(-f(x)\text{,}\) where \(f(x)=-3x^2-7x+1\text{.}\)
To find \(f(-x)\text{,}\) we use an input of \(-x\) in our function \(f\) and simplify to get:
To find \(-f(x)\text{,}\) we take the opposite of the function \(f\) and simplify to get:
Example 11.6.20. Other Nontrivial Simplifications.
If \(g(x)=2x^2-3x-5\text{,}\) find and simplify a formula for \(g(x-1)\text{.}\)
To find \(g(x-1)\text{,}\) we put in \((x-1)\) for the input. It is important to keep the parentheses because we have exponents and negative signs in the function.
Subsection 11.6.5 Technical Definition of a Function
In Section 11.5 we gave a formal definition of a function 11.5.2 and learned to identify what is and is not a function with sets or ordered pairs, tables and graphs. We also used the vertical line test 11.5.20.
Example 11.6.21. Formally Defining a Function.
We learned that sets of ordered pairs, tables and graphs can meet the formal definition of a function. Here is an example that shows a function in all three forms. We can verify that each input has at most one output.
\(\{(1,4),(2,4),(3,3),(4,6),(5,-2)\}\)
| \(x\) | \(f(x)\) |
| \(1\) | \(4\) |
| \(2\) | \(4\) |
| \(3\) | \(3\) |
| \(4\) | \(6\) |
| \(5\) | \(-2\) |
Example 11.6.25. Identifying What is Not a Function.
Identify whether each graph represents a function using the vertical line test 11.5.20.
Exercises 11.6.6 Exercises
Function Basics
1.
Randi will spend \({\$225}\) to purchase some bowls and some plates. Each plate costs \({\$8}\text{,}\) and each bowl costs \({\$5}\text{.}\) The function \(q(x)={-{\frac{8}{5}}x+45}\) models the number of bowls Randi will purchase, where \(x\) represents the number of plates to be purchased.
Interpret the meaning of \(q(35)={-11}\text{.}\)
A. \(-\$11\) will be used to purchase bowls, and \(\$35\) will be used to purchase plates.
B. \(-11\) plates and \(35\) bowls can be purchased.
C. \(35\) plates and \(-11\) bowls can be purchased.
D. \(\$35\) will be used to purchase bowls, and \(-\$11\) will be used to purchase plates.
2.
Douglas will spend \({\$150}\) to purchase some bowls and some plates. Each plate costs \({\$5}\text{,}\) and each bowl costs \({\$6}\text{.}\) The function \(q(x)={-{\frac{5}{6}}x+25}\) models the number of bowls Douglas will purchase, where \(x\) represents the number of plates to be purchased.
Interpret the meaning of \(q(12)={15}\text{.}\)
A. \(\$15\) will be used to purchase bowls, and \(\$12\) will be used to purchase plates.
B. \(12\) plates and \(15\) bowls can be purchased.
C. \(15\) plates and \(12\) bowls can be purchased.
D. \(\$12\) will be used to purchase bowls, and \(\$15\) will be used to purchase plates.
3.
Evaluate the function at the given values.
\(\displaystyle{G(x)={-\frac{9}{x-7}}}\) .
\(\displaystyle{G(8)=}\) .
\(\displaystyle{G(7)=}\) .
4.
Evaluate the function at the given values.
\(\displaystyle{G(x)={\frac{40}{x+8}}}\) .
\(\displaystyle{G(2)=}\) .
\(\displaystyle{G(-8)=}\) .
5.
Use the graph of \(H\) below to evaluate the given expressions. (Estimates are OK.)
\(H(-4)={}\)
\(H(11)={}\)
6.
Use the graph of \(K\) below to evaluate the given expressions. (Estimates are OK.)
\(K(-3)={}\)
\(K(5)={}\)
7.
Use the table of values for \(f\) below to evaluate the given expressions.
| \(x\) | \(-5\) | \(-1\) | \(3\) | \(7\) | \(11\) |
| \(f(x)\) | \(-1.5\) | \(-1.2\) | \(7.3\) | \(6.2\) | \(5.6\) |
\(f({3})={}\)
\(f({7})={}\)
8.
Use the table of values for \(f\) below to evaluate the given expressions.
| \(x\) | \(0\) | \(2\) | \(4\) | \(6\) | \(8\) |
| \(f(x)\) | \(7.3\) | \(-1.3\) | \(1.3\) | \(7.3\) | \(-1.6\) |
\(f({0})={}\)
\(f({8})={}\)
9.
Make a table of values for the function \(h\text{,}\) defined by \(h(x)={-4x^{2}}\text{.}\) Based on values in the table, sketch a graph of \(h\text{.}\)
| \(x\) | \(h(x)\) |
10.
Make a table of values for the function \(H\text{,}\) defined by \(\displaystyle H(x)={\frac{2^{x}-3}{x^{2}+3}}\text{.}\) Based on values in the table, sketch a graph of \(H\text{.}\)
| \(x\) | \(H(x)\) |
11.
The following figure has the graph \(y=d(t)\text{,}\) which models a particleâs distance from the starting line in feet, where \(t\) stands for time in seconds since timing started.
Find \(d(7)\text{.}\)
-
Interpret the meaning of \(d(7)\text{.}\)
A. The particle was \(9\) feet away from the starting line \(7\) seconds since timing started.
B. The particle was \(7\) feet away from the starting line \(9\) seconds since timing started.
C. In the first \(9\) seconds, the particle moved a total of \(7\) feet.
D. In the first \(7\) seconds, the particle moved a total of \(9\) feet.
Solve \(d(t)={3}\) for \(t\text{.}\) \(t=\)
-
Interpret the meaning of part câs solution(s).
A. The particle was \(3\) feet from the starting line \(1\) seconds since timing started, and again \(9\) seconds since timing started.
B. The particle was \(3\) feet from the starting line \(9\) seconds since timing started.
C. The particle was \(3\) feet from the starting line \(1\) seconds since timing started.
D. The particle was \(3\) feet from the starting line \(1\) seconds since timing started, or \(9\) seconds since timing started.
12.
The following figure has the graph \(y=d(t)\text{,}\) which models a particleâs distance from the starting line in feet, where \(t\) stands for time in seconds since timing started.
Find \(d(8)\text{.}\)
-
Interpret the meaning of \(d(8)\text{.}\)
A. In the first \(2\) seconds, the particle moved a total of \(8\) feet.
B. The particle was \(8\) feet away from the starting line \(2\) seconds since timing started.
C. In the first \(8\) seconds, the particle moved a total of \(2\) feet.
D. The particle was \(2\) feet away from the starting line \(8\) seconds since timing started.
Solve \(d(t)={3}\) for \(t\text{.}\) \(t=\)
-
Interpret the meaning of part câs solution(s).
A. The particle was \(3\) feet from the starting line \(3\) seconds since timing started, or \(7\) seconds since timing started.
B. The particle was \(3\) feet from the starting line \(3\) seconds since timing started.
C. The particle was \(3\) feet from the starting line \(7\) seconds since timing started.
D. The particle was \(3\) feet from the starting line \(3\) seconds since timing started, and again \(7\) seconds since timing started.
Domain and Range
13.
A function is graphed.
This function has domain and range .
14.
A function is graphed.
This function has domain and range .
15.
A function is graphed.
The function has domain and range .
16.
A function is graphed.
The function has domain and range .
17.
A function is graphed.
The function has domain and range .
18.
A function is graphed.
The function has domain and range .
19.
Find the domain of \(t\) where \(\displaystyle{t(x)= \frac{\sqrt{8 + x}}{5 - x}}\text{.}\)
20.
Find the domain of \(C\) where \(\displaystyle{C(x)= \frac{\sqrt{10 + x}}{2 - x}}\text{.}\)
21.
An object was shot up into the air at an initial vertical speed of \(512\) feet per second. Its height as time passes can be modeled by the quadratic function \(f\text{,}\) where \(f(t)={-16t^{2}+512t}\text{.}\) Here \(t\) represents the number of seconds since the objectâs release, and \(f(t)\) represents the objectâs height in feet.
Find the functionâs domain and range in this context.
The functionâs domain in this context is .
The functionâs range in this context is .
22.
An object was shot up into the air at an initial vertical speed of \(544\) feet per second. Its height as time passes can be modeled by the quadratic function \(f\text{,}\) where \(f(t)={-16t^{2}+544t}\text{.}\) Here \(t\) represents the number of seconds since the objectâs release, and \(f(t)\) represents the objectâs height in feet.
Find the functionâs domain and range in this context.
The functionâs domain in this context is .
The functionâs range in this context is .
Using Technology to Explore Functions
23.
Use technology to make a table of values for the function \(H\) defined by \(H(x)={-4x^{2}+16x+1}\text{.}\)
| \(x\) | \(H(x)\) |
24.
Use technology to make a table of values for the function \(K\) defined by \(K(x)={2x^{2}-7x-2}\text{.}\)
| \(x\) | \(K(x)\) |
25.
Choose an appropriate window for graphing the function \(f\) defined by \(f(x)={-1184x-7607}\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
26.
Choose an appropriate window for graphing the function \(f\) defined by \(f(x)={-139x+159}\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
27.
Use technology to determine how many times the equations \(y={-5x^{3}+2x^{2}+x}\) and \(y={6x+4}\) intersect. They intersect
zero times
one time
two times
three times
28.
Use technology to determine how many times the equations \(y={-3x^{3}-x^{2}+9x}\) and \(y={-5x+6}\) intersect. They intersect
zero times
one time
two times
three times
29.
For the function \(L\) defined by
use technology to determine the following. Round answers as necessary.
Any intercepts.
The vertex.
The domain.
The range.
30.
For the function \(M\) defined by
use technology to determine the following. Round answers as necessary.
Any intercepts.
The vertex.
The domain.
The range.
31.
Let \(f(x)=4x^2+5x-1\) and \(g(x)=5\text{.}\) Use graphing technology to determine the following.
What are the points of intersection for these two functions?
Solve \(f(x)=g(x)\text{.}\)
Solve \(f(x)\lt g(x)\text{.}\)
Solve \(f(x)\geq g(x)\text{.}\)
32.
Let \(p(x)=6x^2-3x+4\) and \(k(x)=7\text{.}\) Use graphing technology to determine the following.
What are the points of intersection for these two functions?
Solve \(p(x)=k(x)\text{.}\)
Solve \(p(x)\lt k(x)\text{.}\)
Solve \(p(x)\geq k(x)\text{.}\)
33.
Use graphing technology to solve the equation \(-0.02 x^2 + 1.97 x - 51.5=0.05\left(x-50\right)^2-0.03\left(x-50\right)\text{.}\) Approximate the solution(s) if necessary.
34.
Use graphing technology to solve the equation \(-200x^2+60x-55=-20x-40\text{.}\) Approximate the solution(s) if necessary.
35.
Use graphing technology to solve the inequality \(-15x^2-6\leq 10x-4\text{.}\) State the solution set using interval notation, and approximate if necessary.
36.
Use graphing technology to solve the inequality \(\frac{1}{2}x^2+\frac{3}{2}x \geq \frac{1}{2}x-\frac{3}{2}\text{.}\) State the solution set using interval notation, and approximate if necessary.
Simplifying Expressions with Function Notation
37.
Let \(f\) be a function given by \(f(x)={-3x^{2}+3x}\text{.}\) Find and simplify the following:
\(f(x) - 3={}\)
\(f(x - 3)={}\)
\(-3f(x)={}\)
\(f(-3x)={}\)
38.
Let \(f\) be a function given by \(f(x)={3x^{2}-4x}\text{.}\) Find and simplify the following:
\(f(x) - 4={}\)
\(f(x - 4)={}\)
\(-4f(x)={}\)
\(f(-4x)={}\)
39.
Simplify \(F(r)+6\text{,}\) where \(F(r)={3-5.1r}\text{.}\)
40.
Simplify \(g(r)+9\text{,}\) where \(g(r)={2+6.5r}\text{.}\)
Technical Definition of a Function
41.
Does the following set of ordered pairs make for a function of \(x\text{?}\)
\(\Big\{(-3,3),(-5,9),(-5,0),(1,7),(-6,3)\Big\}\)
This set of ordered pairs
describes
does not describe
42.
Does the following set of ordered pairs make for a function of \(x\text{?}\)
\(\Big\{(-8,9),(5,5),(-3,9),(-1,2),(-8,10)\Big\}\)
This set of ordered pairs
describes
does not describe
43.
Below is all of the information that exists about a function \(f\text{.}\)
\(\begin{aligned} f(-2)\amp =1\amp f(0)\amp =5\amp f(1)\amp =2 \end{aligned}\)
Write \(f\) as a set of ordered pairs.
\(f\) has domain and range .
44.
Below is all of the information about a function \(f\text{.}\)
\(\begin{aligned} f(a)\amp =5\amp f(b)\amp =6\\ f(c)\amp =3\amp f(d)\amp =6 \end{aligned}\)
Write \(f\) as a set of ordered pairs.
\(f\) has domain and range .
45.
The following graphs show two relationships. Decide whether each graph shows a relationship where \(y\) is a function of \(x\text{.}\)
The first graph
does
does not
does
does not
46.
The following graphs show two relationships. Decide whether each graph shows a relationship where \(y\) is a function of \(x\text{.}\)
The first graph
does
does not
does
does not
47.
Some equations involving \(x\) and \(y\) define \(y\) as a function of \(x\text{,}\) and others do not. For example, if \(x+y=1\text{,}\) we can solve for \(y\) and obtain \(y = 1-x\text{.}\) And we can then think of \(y = f(x) = 1-x\text{.}\) On the other hand, if we have the equation \(x=y^2\) then \(y\) is not a function of \(x\text{,}\) since for a given positive value of \(x\text{,}\) the value of \(y\) could equal \(\sqrt{x}\) or it could equal \(-\sqrt{x}\text{.}\) Select all of the following relations that make \(y\) a function of \(x\text{.}\) There are several correct answers.
\(y^3 + x^4 = 1\)
\(y + x^2 = 1\)
\(y^2 + x^2 = 1\)
\(y^6 + x = 1\)
\(y - \left|x\right| = 0\)
\(5 x + 8 y + 9 = 0\)
\(x+y=1\)
\(\left|y\right| - x = 0\)
On the other hand, some equations involving \(x\) and \(y\) define \(x\) as a function of \(y\) (the other way round).
Select all of the following relations that make \(x\) a function of \(y\text{.}\) There are several correct answers.
\(\left|y\right| - x = 0\)
\(5 x + 8 y + 9 = 0\)
\(y^2 + x^2 = 1\)
\(y - \left|x\right| = 0\)
\(y^4 + x^5 = 1\)
48.
Some equations involving \(x\) and \(y\) define \(y\) as a function of \(x\text{,}\) and others do not. For example, if \(x+y=1\text{,}\) we can solve for \(y\) and obtain \(y = 1-x\text{.}\) And we can then think of \(y = f(x) = 1-x\text{.}\) On the other hand, if we have the equation \(x=y^2\) then \(y\) is not a function of \(x\text{,}\) since for a given positive value of \(x\text{,}\) the value of \(y\) could equal \(\sqrt{x}\) or it could equal \(-\sqrt{x}\text{.}\) Select all of the following relations that make \(y\) a function of \(x\text{.}\) There are several correct answers.
\(\left|y\right| - x = 0\)
\(y + x^2 = 1\)
\(x+y=1\)
\(y^2 + x^2 = 1\)
\(6 x + 5 y + 4 = 0\)
\(y^3 + x^4 = 1\)
\(y^6 + x = 1\)
\(y - \left|x\right| = 0\)
On the other hand, some equations involving \(x\) and \(y\) define \(x\) as a function of \(y\) (the other way round).
Select all of the following relations that make \(x\) a function of \(y\text{.}\) There are several correct answers.
\(y^2 + x^2 = 1\)
\(\left|y\right| - x = 0\)
\(y - \left|x\right| = 0\)
\(y^4 + x^5 = 1\)
\(6 x + 5 y + 4 = 0\)
49.
Determine whether or not the following table could be the table of values of a function. If the table can not be the table of values of a function, give an input that has more than one possible output.
| Input | Output |
| \(2\) | \(7\) |
| \(4\) | \(-18\) |
| \(6\) | \(10\) |
| \(8\) | \(-4\) |
| \(-2\) | \(1\) |
Could this be the table of values for a function?
yes
no
If not, which input has more than one possible output?
-2
2
4
6
8
None, the table represents a function.
50.
Determine whether or not the following table could be the table of values of a function. If the table can not be the table of values of a function, give an input that has more than one possible output.
| Input | Output |
| \(2\) | \(12\) |
| \(4\) | \(9\) |
| \(6\) | \(-14\) |
| \(8\) | \(-15\) |
| \(-2\) | \(1\) |
Could this be the table of values for a function?
yes
no
If not, which input has more than one possible output?
-2
2
4
6
8
None, the table represents a function.
51.
Determine whether or not the following table could be the table of values of a function. If the table can not be the table of values of a function, give an input that has more than one possible output.
| Input | Output |
| \(-4\) | \(6\) |
| \(-3\) | \(2\) |
| \(-2\) | \(0\) |
| \(-3\) | \(13\) |
| \(-1\) | \(-3\) |
Could this be the table of values for a function?
yes
no
If not, which input has more than one possible output?
-4
-3
-2
-1
None, the table represents a function.
52.
Determine whether or not the following table could be the table of values of a function. If the table can not be the table of values of a function, give an input that has more than one possible output.
| Input | Output |
| \(-4\) | \(-14\) |
| \(-3\) | \(4\) |
| \(-2\) | \(17\) |
| \(-3\) | \(19\) |
| \(-1\) | \(15\) |
Could this be the table of values for a function?
yes
no
If not, which input has more than one possible output?
-4
-3
-2
-1
None, the table represents a function.