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Section10.4Simplifying Expressions with Function Notation

ObjectivesPCC Course Content and Outcome Guide
Figure10.4.1Alternative Video Lesson

In this section, we will discuss algebra simplification that will appear in many facets of education. Simplification is a skill, like cooking noodles or painting a wall. It may not always be exciting, but it does serve a purpose. Also like cooking noodles or painting a wall, it isn't usually difficult, and yet there are common avoidable mistakes that people make. With practice from this section, you'll have experience to prevent yourself from overcooking the noodles or ruining your paintbrush.

Subsection10.4.1Negative Signs in and out of Function Notation

Let's start by reminding ourselves about the meaning of function notation. When we write \(f(x)\text{,}\) we have a process \(f\) that is doing something to an input value \(x\text{.}\) Whatever is inside those parentheses is the input to the function. What if we use something for input that is not quite as simple as “\(x\)”?

Example10.4.2

Find and simplify a formula for \(f(-x)\text{,}\) where \(f(x)=x^2+3x-4\text{.}\)

Solution

Those parentheses encase “\(-x\)”, so we are meant to treat “\(-x\)” as the input. The rule that we have been given for \(f\) is

\begin{equation*} f(x)=x^2+3x-4\text{.} \end{equation*}

But the \(x\)'s that are in this formula are just place-holders. What \(f\) does to a number can just as easily be communicated with

\begin{equation*} f(\phantom{x})=(\phantom{x})^2+3(\phantom{x})-4\text{.} \end{equation*}

So now that we are meant to treat “\(-x\)” as the input, we will insert “\(-x\)” into those slots, after which we can do more familiar algebraic simplification:

\begin{align*} f(\phantom{-x})\amp=(\phantom{-x})^2+3(\phantom{-x})-4\\ f(\substitute{-x})\amp=(\substitute{-x})^2+3(\substitute{-x})-4\\ \amp=x^2-3x-4 \end{align*}

The previous example contrasts nicely with this one:

Example10.4.3

Find and simplify a formula for \(-f(x)\text{,}\) where \(f(x)=x^2+3x-4\text{.}\)

Solution

Here, the parentheses only encase “\(x\)”. The negative sign is on the outside. So the way to see this expression is that first \(f\) will do what it does to \(x\text{,}\) and then that result will be negated:

\begin{align*} -f(x)\amp=-(\substitute{x^2+3x-4})\\ \amp=-x^2-3x+4 \end{align*}

Note that the answer to this exercise, which was to simplify \(-f(x)\text{,}\) is different from the answer to Exercise 10.4.2, which was to simplify \(f(-x)\text{.}\) In general you cannot pass a negative sign in and out of function notation and still have the same quantity.

In Example 10.4.2 and Example 10.4.3, we are working with the expressions \(f(-x)\) and \(-f(x)\text{,}\) and trying to find “simplified” formulas. If it seems strange to be doing these things, perhaps this applied example will help.

Checkpoint10.4.4

Subsection10.4.2Other Nontrivial Simplifications

Example10.4.5

Find and simplify a formula for \(h(5x)\text{,}\) where \(h(x)=\frac{x}{x-2}\text{.}\)

Solution

The parentheses encase “\(5x\)”, so we are meant to treat “\(5x\)” as the input.

\begin{align*} h(\phantom{5x})\amp=\frac{\substitute{(\phantom{5x})}}{\substitute{(\phantom{5x})}-2}\\ h(\substitute{5x})\amp=\frac{\substitute{5x}}{\substitute{5x}-2}\\ \amp=\frac{5x}{5x-2} \end{align*}
Example10.4.6

Find and simplify a formula for \(\frac{1}{3}g(3x)\text{,}\) where \(g(x)=2x^2+8\text{.}\)

Solution

Do the \(\frac{1}{3}\) and the \(3\) cancel each other? No. The \(3\) is part of the input, affecting \(x\) right away. Then \(g\) does whatever it does to \(3x\text{,}\) and then we multiply the result by \(\frac{1}{3}\text{.}\) Since the function \(g\) acts “in between,” we don't have the chance to cancel the \(3\) with the \(\frac{1}{3}\text{.}\) Let's see what actually happens:

Those parentheses encase “\(3x\)”, so we are meant to treat “\(3x\)” as the input. We will keep the \(\frac{1}{3}\) where it is until it is possible to simplify:

\begin{align*} \frac{1}{3}g(\phantom{3x})\amp=\frac{1}{3}\left(2(\phantom{3x})^2+8\right)\\ \frac{1}{3}g(\substitute{3x})\amp=\frac{1}{3}\left(2(\substitute{3x})^2+8\right)\\ \amp=\frac{1}{3}\left(2\left(9x^2\right)+8\right)\\ \amp=\frac{1}{3}\left(18x^2+8\right)\\ \amp=6x^2+\frac{8}{3} \end{align*}
Example10.4.7

If \(k(x)=x^2-3x\text{,}\) find and simplify a formula for \(k(x-4)\text{.}\)

Solution

This type of exercise is often challenging for algebra students. But let's focus on those parentheses one more time. They encase “\(x-4\)”, so we are meant to treat “\(x-4\)” as the input.

\begin{align*} k(\phantom{x-4})\amp=(\phantom{x-4})^2-3(\phantom{x-4})\\ k(\substitute{x-4})\amp=(\substitute{x-4})^2-3(\substitute{x-4})\\ \amp=x^2-8x+16-3x+12\\ \amp=x^2-11x+28 \end{align*}
Checkpoint10.4.8
Example10.4.9

If \(f(x)=\frac{1}{x}\text{,}\) find and simplify a formula for \(f(x+3)+2\text{.}\)

Solution

Do not be tempted to add the \(3\) and the \(2\text{.}\) The \(3\) is added to input before the function \(f\) does its work. The \(2\) is added to the result after \(f\) has done its work.

\begin{align*} f(\phantom{x+3})+2\amp=\frac{1}{\substitute{(\phantom{x+3})}}+2\\ f(\substitute{x+3})+2\amp=\frac{1}{\substitute{x+3}}+2 \end{align*}

This last expression is considered fully simplified. However you might combine the two terms using a technique from Section 13.3.

The tasks we have practiced in this section are the kind of tasks that will make it easier to understand interesting and useful material in college algebra and calculus.

SubsectionExercises

Find simplified formulas for these expressions.

1

Simplify \(K({r+7})\text{,}\) where \(K(r)={-5+8r}\text{.}\)

2

Simplify \(F({r+1})\text{,}\) where \(F(r)={-5+3r}\text{.}\)

3

Simplify \(g({-r})\text{,}\) where \(g(r)={-5-2r}\text{.}\)

4

Simplify \(H({-t})\text{,}\) where \(H(t)={-6+7t}\text{.}\)

5

Simplify \(F({t+4})\text{,}\) where \(F(t)={-6+5.6t}\text{.}\)

6

Simplify \(f({x+7})\text{,}\) where \(f(x)={-6+1.1x}\text{.}\)

7

Simplify \(H({x - {\frac{3}{2}}})\text{,}\) where \(H(x)={-{\frac{8}{9}} - {\frac{7}{6}}x}\text{.}\)

8

Simplify \(F({y - {\frac{7}{2}}})\text{,}\) where \(F(y)={{\frac{3}{2}} - {\frac{6}{5}}y}\text{.}\)

9

Simplify \(f(y)+1\text{,}\) where \(f(y)={4y-7}\text{.}\)

10

Simplify \(H(r)+4\text{,}\) where \(H(r)={-r-7}\text{.}\)

11

Simplify \(h(r)+8\text{,}\) where \(h(r)={-7-5.1r}\text{.}\)

12

Simplify \(f(r)+3\text{,}\) where \(f(r)={-7+6.5r}\text{.}\)

13

Simplify \(G({7t})\text{,}\) where \(G(t)={2t^{2}-7t+2}\text{.}\)

14

Simplify \(h({2t})\text{,}\) where \(h(t)={-3t^{2}-8t-7}\text{.}\)

15

Simplify \(f({-x})\text{,}\) where \(f(x)={-7x^{2}-8x+1}\text{.}\)

16

Simplify \(G({-x})\text{,}\) where \(G(x)={7x^{2}+5x-8}\text{.}\)

17

Simplify \(4h(y)\text{,}\) where \(h(y)={-8y^{2}+y+8}\text{.}\)

18

Simplify \(8K(y)\text{,}\) where \(K(y)={-4y^{2}+8y-7}\text{.}\)

19

Simplify \(G({r-6})\text{,}\) where \(G(r)={7.5r^{2}+8r+1}\text{.}\)

20

Simplify \(g({r+2})\text{,}\) where \(g(r)={3.1r^{2}+8r-7}\text{.}\)

21

Simplify \(K(r)+2\text{,}\) where \(K(r)={-r^{2}+8r+1}\text{.}\)

22

Simplify \(F(t)+5\text{,}\) where \(F(t)={-6t^{2}+8t-7}\text{.}\)

23

Simplify \(K({x+2})\text{,}\) where \(K(x)={\sqrt{7+7x}}\text{.}\)

24

Simplify \(f({x+8})\text{,}\) where \(f(x)={\sqrt{7+2x}}\text{.}\)

25

Simplify \(g(x)+5\text{,}\) where \(g(x)={\sqrt{7-6x}}\text{.}\)

26

Simplify \(g(x)+2\text{,}\) where \(g(x)={\sqrt{7-x}}\text{.}\)

27

Simplify \(h({x+8})\text{,}\) where \(h(x)={x+\sqrt{6+5x}}\text{.}\)

28

Simplify \(F({x+5})\text{,}\) where \(F(x)={-9x+\sqrt{6-7x}}\text{.}\)

29

Simplify \(g({s+4})\text{,}\) where \(g(s)={\frac{1}{-5s+6}}\text{.}\)

30

Simplify \(G({s+7})\text{,}\) where \(G(s)={-\frac{9}{7s+6}}\text{.}\)

31

Simplify \(F({3t})\text{,}\) where \(F(t)={\frac{2t}{6t^{2}+1}}\text{.}\)

32

Simplify \(g({-6t})\text{,}\) where \(g(t)={\frac{3t}{5t^{2}-8}}\text{.}\)

33

Let \(f\) be a function given by \(f(x)={2x-9}\text{.}\) Find and simplify the following:

  1. \(f(x)+2={}\)

  2. \(f(x+2)={}\)

  3. \(2f(x)={}\)

  4. \(f(2x)={}\)

34

Let \(f\) be a function given by \(f(x)={-3x-1}\text{.}\) Find and simplify the following:

  1. \(f(x)+8={}\)

  2. \(f(x+8)={}\)

  3. \(8f(x)={}\)

  4. \(f(8x)={}\)

35

Let \(f\) be a function given by \(f(x)={-3x^{2}+4x}\text{.}\) Find and simplify the following:

  1. \(f(x) - 3={}\)

  2. \(f(x - 3)={}\)

  3. \(-3f(x)={}\)

  4. \(f(-3x)={}\)

36

Let \(f\) be a function given by \(f(x)={3x^{2}-2x}\text{.}\) Find and simplify the following:

  1. \(f(x) - 4={}\)

  2. \(f(x - 4)={}\)

  3. \(-4f(x)={}\)

  4. \(f(-4x)={}\)

37

A circular oil slick is expanding with radius, \(r\) in feet, at time \(t\) in hours given by \(r= 12 t - 0.3 t^2,\) for \(t\) in hours, \(0 \le t \le 10.\)

Find a formula for \(A = f(t)\text{,}\) the area of the oil slick as a function of time.

\(A = f(t) =\)

with

(Be sure to include units!)

38

Suppose \(T(t)\) represents the temperature outside, in Fahrenheit, at \(t\) hours past noon, and a formula for \(T\) is \(T(t)={\frac{20t}{t^{2}+1}+52}\text{.}\)

If we introduce \(F(t)\) as the temperature outside, in Fahrenheit, at \(t\) hours past 1:00pm, then \(F(t)=T(t+1)\text{.}\) Find a simplified formula for \(T(t+1)\text{.}\)

\(T(t+1) =\)

39

Suppose \(G(t)\) represents how many gigabytes of data has been downloaded \(t\) minutes after you started a download.

If we introduce \(M(t)\) as how many megabytes of data has been downloaded \(t\) minutes after you started a download, then \(M(t)=1024G(t)\text{.}\) Find a simplified formula for \(1024G(t)\text{.}\)

\(1024G(t) =\)