Objectives: PCC Course Content and Outcome Guide
[cross-reference to target(s) "ccog-distinguish-and-simplify-simple-function-composition" missing or not unique]
Section 11.4 Simplifying Expressions with Function Notation
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.
Subsection 11.4.1 Negative 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 “\(x\text{?}\)”
Example 11.4.2.
Find and simplify a formula for \(f(-x)\text{,}\) where \(f(x)=x^2+3x-4\text{.}\)
Explanation.
Those parentheses encase “\(-x\text{,}\)” 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:
Example 11.4.3.
Find and simplify a formula for \(-f(x)\text{,}\) where \(f(x)=x^2+3x-4\text{.}\)
Explanation.
Here, the parentheses only encase “\(x\text{.}\)” 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 Example 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 2 and Example 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.
Checkpoint 11.4.4.
The NASDAQ Composite Index measures how well a portion of the stock market is doing. Suppose \(N(t)\) is the value of the index \(t\) days after January 1, 2018. A formula for \(N\) is \(N(t)={3.34t^{2}+26.2t+6980}\text{.}\)
What if you wanted a new function, \(B\text{,}\) that gives the value of the NASDAQ index \(t\) days before January 1, 2018? Technically, \(t\) days before is the same as negative \(t\) days after. So \(B(t)\) is the same as \(N(-t)\text{,}\) and now the expression \(N(-t)\) means something. Find a simplified formula for \(N(-t)\text{.}\)
\(N(-t)=\)
Explanation.
\begin{equation*}
\begin{aligned}
N(\phantom{-t})\amp=3.34(\phantom{-t})^2+26.2(\phantom{-t})+6980\\
N(\substitute{-t})\amp=3.34(\substitute{-t})^2+26.2(\substitute{-t})+6980\\
\amp=3.34t^2-26.2t+6980
\end{aligned}
\end{equation*}
Subsection 11.4.2 Other Nontrivial Simplifications
Example 11.4.5.
Find and simplify a formula for \(h(5x)\text{,}\) where \(h(x)=\frac{x}{x-2}\text{.}\)
Explanation.
The parentheses encase “\(5x\text{,}\)” 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*}
Example 11.4.6.
Find and simplify a formula for \(\frac{1}{3}g(3x)\text{,}\) where \(g(x)=2x^2+8\text{.}\)
Explanation.
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 “\(3x\text{,}\)” 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*}
Example 11.4.7.
If \(k(x)=x^2-3x\text{,}\) find and simplify a formula for \(k(x-4)\text{.}\)
Explanation.
This type of exercise is often challenging for algebra students. But let’s focus on those parentheses one more time. They encase “\(x-4\text{,}\)” 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*}
Checkpoint 11.4.8.
If \(q(x)=x+\sqrt{x+8}\text{,}\) find and simplify a formula for \(q(x+5)\text{.}\)
\(q(x+5)=\)
Explanation.
Starting with the generic formula for \(q\text{:}\)
\begin{equation*}
\begin{aligned}
q(\phantom{x+5})\amp=(\phantom{x+5})+\sqrt{(\phantom{x+5})+8}\\
q(\substitute{x+5})\amp=\substitute{x+5}+\sqrt{\substitute{x+5}+8}\\
\amp=x+5+\sqrt{x+13}
\end{aligned}
\end{equation*}
Example 11.4.9.
If \(f(x)=\frac{1}{x}\text{,}\) find and simplify a formula for \(f(x+3)+2\text{.}\)
Explanation.
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 12.3.
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.
Reading Questions 11.4.3 Reading Questions
1.
Explain how \(f(x+2)\) probably does not mean that \(f\) is being multiplied by \(x+2\text{.}\)
Exercises 11.4.4 Exercises
Review and Warmup.
1.
Use the distributive property to write an equivalent expression to \({3\mathopen{}\left(c+7\right)}\) that has no grouping symbols.
2.
Use the distributive property to write an equivalent expression to \({2\mathopen{}\left(n+6\right)}\) that has no grouping symbols.
3.
Use the distributive property to write an equivalent expression to \({-6\mathopen{}\left(c-6\right)}\) that has no grouping symbols.
4.
Use the distributive property to write an equivalent expression to \({-9\mathopen{}\left(n+9\right)}\) that has no grouping symbols.
5.
Multiply the polynomials.
\(3\left({t+8}\right)^2=\)
6.
Multiply the polynomials.
\(6\left({x+5}\right)^2=\)
7.
Expand the square of a binomial.
\(\left({2x+7}\right)^2=\)
8.
Expand the square of a binomial.
\(\left({8y+6}\right)^2=\)
Simplifying Function Expressions.
9.
Simplify \(h({y+9})\text{,}\) where \(h(y)={2+y}\text{.}\)
10.
Simplify \(f({y+4})\text{,}\) where \(f(y)={2-4y}\text{.}\)
11.
Simplify \(G({-r})\text{,}\) where \(G(r)={8+6r}\text{.}\)
12.
Simplify \(h({-r})\text{,}\) where \(h(r)={4-4r}\text{.}\)
13.
Simplify \(K({t+6})\text{,}\) where \(K(t)={1-0.9t}\text{.}\)
14.
Simplify \(G({t+1})\text{,}\) where \(G(t)={1-5.3t}\text{.}\)
15.
Simplify \(h({x+{\frac{7}{6}}})\text{,}\) where \(h(x)={-{\frac{3}{2}} - {\frac{2}{9}}x}\text{.}\)
16.
Simplify \(K({x+{\frac{4}{3}}})\text{,}\) where \(K(x)={{\frac{9}{5}}+{\frac{1}{9}}x}\text{.}\)
17.
Simplify \(G(x)+3\text{,}\) where \(G(x)={-3x+1}\text{.}\)
18.
Simplify \(g(y)+7\text{,}\) where \(g(y)={-7y+6}\text{.}\)
19.
Simplify \(K(y)+2\text{,}\) where \(K(y)={-2+4.5y}\text{.}\)
20.
Simplify \(F(r)+6\text{,}\) where \(F(r)={6+0.1r}\text{.}\)
21.
Simplify \(g({9r})\text{,}\) where \(g(r)={-5r^{2}-2r-1}\text{.}\)
22.
Simplify \(H({5t})\text{,}\) where \(H(t)={8t^{2}-t+6}\text{.}\)
23.
Simplify \(F({-t})\text{,}\) where \(F(t)={6t^{2}+3t-1}\text{.}\)
24.
Simplify \(g({-x})\text{,}\) where \(g(x)={-4x^{2}-2x-1}\text{.}\)
25.
Simplify \(7H(x)\text{,}\) where \(H(x)={-6x^{2}-x-3}\text{.}\)
26.
Simplify \(2F(x)\text{,}\) where \(F(x)={6x^{2}-x+6}\text{.}\)
27.
Simplify \(f({y+1})\text{,}\) where \(f(y)={-1.6y^{2}-3y+8}\text{.}\)
28.
Simplify \(H({y+8})\text{,}\) where \(H(y)={-3.4y^{2}-2y+6}\text{.}\)
29.
Simplify \(h(r)+4\text{,}\) where \(h(r)={-8r^{2}-2r-3}\text{.}\)
30.
Simplify \(f(r)+7\text{,}\) where \(f(r)={4r^{2}-2r+6}\text{.}\)
31.
Simplify \(H({x+7})\text{,}\) where \(H(x)={\sqrt{-3-x}}\text{.}\)
32.
Simplify \(K({x+4})\text{,}\) where \(K(x)={\sqrt{-3-5x}}\text{.}\)
33.
Simplify \(f(x)+1\text{,}\) where \(f(x)={\sqrt{-3+7x}}\text{.}\)
34.
Simplify \(f(x)+7\text{,}\) where \(f(x)={\sqrt{-3+3x}}\text{.}\)
35.
Simplify \(g({x+4})\text{,}\) where \(g(x)={-3x+\sqrt{-3-6x}}\text{.}\)
36.
Simplify \(h({x+9})\text{,}\) where \(h(x)={6x+\sqrt{-4-2x}}\text{.}\)
37.
Simplify \(G({z+6})\text{,}\) where \(G(z)={-\frac{3}{5z-4}}\text{.}\)
38.
Simplify \(h({z+1})\text{,}\) where \(h(z)={\frac{8}{-4z+5}}\text{.}\)
39.
Simplify \(K({-6s})\text{,}\) where \(K(s)={\frac{s}{-4s^{2}-3}}\text{.}\)
40.
Simplify \(G({9s})\text{,}\) where \(G(s)={\frac{2s}{-5s^{2}+5}}\text{.}\)
Exercise Group.
41.
Let \(f\) be a function given by \(f(x)={-9x-3}\text{.}\) Find and simplify the following:
- \(f(x)+7={}\)
- \(f(x+7)={}\)
- \(7f(x)={}\)
- \(f(7x)={}\)
42.
Let \(f\) be a function given by \(f(x)={2x+5}\text{.}\) Find and simplify the following:
- \(f(x)+5={}\)
- \(f(x+5)={}\)
- \(5f(x)={}\)
- \(f(5x)={}\)
43.
Let \(f\) be a function given by \(f(x)={2x^{2}-4x}\text{.}\) Find and simplify the following:
- \(f(x) - 5={}\)
- \(f(x - 5)={}\)
- \(-5f(x)={}\)
- \(f(-5x)={}\)
44.
Let \(f\) be a function given by \(f(x)={-2x^{2}+x}\text{.}\) Find and simplify the following:
- \(f(x) - 2={}\)
- \(f(x - 2)={}\)
- \(-2f(x)={}\)
- \(f(-2x)={}\)
Applications.
45.
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!)
46.
Suppose \(T(t)\) represents the temperature outside, in Fahrenheit, at \(t\) hours past noon, and a formula for \(T\) is \(T(t)={\frac{25t}{t^{2}+1}+50}\text{.}\)
If we introduce \(F(t)\) as the temperature outside, in Fahrenheit, at \(t\) hours past 2:00pm, then \(F(t)=T(t+2)\text{.}\) Find a simplified formula for \(T(t+2)\text{.}\)
\(T(t+2) =\)
47.
Suppose \(G(t)\) represents how many gigabytes of data has been downloaded \(t\) minutes after you started a download, and a formula for \(G\) is \(G(t)={25-\frac{50}{t+2}}\text{.}\)
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 \(1024G(t)\text{.}\)
\(1024G(t) =\)
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