TBIL-Precal Combining and Composing Functions (FN5)

In Activity 2.5.1 and Activity 2.5.2, we have found the sum, difference, product, and quotient of two functions. We can use the following notation for these newly created functions:

\begin{align*} (f+g)(x) \amp = f(x)+g(x) \\ (f-g)(x) \amp = f(x)-g(x) \\ (f\cdot g)(x) \amp = f(x)\cdot g(x) \\ \left(\dfrac{f}{g}\right)(x) \amp = \dfrac{f(x)}{g(x)} \end{align*}

With $\left(\dfrac{f}{g}\right)(x) \text{,}$ we note that the quotient is only defined when $g(x)\neq 0\text{.}$

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Activity 2.5.4.

Let $\displaystyle f(x)=\dfrac{1}{3x-5}\text{.}$

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(a)

Find $f(4)\text{.}$

$\displaystyle \dfrac{4}{3x-5}$

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$\displaystyle \dfrac{1}{4(3x-5)}$

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$\displaystyle \dfrac{1}{7}$

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$\displaystyle 7$

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Hint.

See Remark 2.2.5 for a reminder of what this notation means!

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Answer.

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(b)

If you were asked to find $f(x^3-2)\text{,}$ how do you think you would proceed?

Multiply the original function $\dfrac{1}{3x-5}$ by $x^3-2\text{.}$

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Plug the expression $x^3-2$ in for all the $x$-values in $\dfrac{1}{3x-5}\text{.}$

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Plug the original function $\dfrac{1}{3x-5}$ in for all the $x$-values in $x^3-2\text{.}$

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Multiply $3x-5$ by $x^3-2\text{.}$

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Answer.

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(c)

Find $f(x^3-2)\text{.}$

$\displaystyle \dfrac{1}{3x-5} \cdot (x^3-2)$

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$\displaystyle \dfrac{1}{3(x^3-2)-5}$

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$\displaystyle \left(\dfrac{1}{3x-5} \right)^3-2$

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$\displaystyle (3x-5)(x^3-2)$

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Answer.

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(d)

What if we gave the expression $x^3-2$ a name? Let’s define $g(x)=x^3-2\text{.}$ What’s another way we could denote $f(x^3-2)\text{?}$

$\displaystyle f(x) \cdot g(x)$

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$\displaystyle g(f(x))$

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$\displaystyle f(g(x))$

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$\displaystyle \dfrac{f(x)}{g(x)} $

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Answer.

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Definition 2.5.5.

Given the functions $f(x)$ and $g(x)\text{,}$ we define the composition of $f$ and $g$ to be the new function $h(x)$ given by

\begin{equation*} h(x) = f(g(x))\text{.} \end{equation*}

We also sometimes use the notation

\begin{equation*} f \circ g \end{equation*}

\begin{equation*} (f \circ g)(x) \end{equation*}

to refer to $f(g(x))\text{.}$

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Remark 2.5.6.

When discussing the composite function $f(g(x))\text{,}$ also written as $(f\circ g)(x)\text{,}$ we often call $g(x)$ the "inner function" and $f(x)$ the "outer function". It is important to note that the inner function is actually the first function that gets applied to a given input, and then the outer function is applied to the output of the inner function.

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Activity 2.5.7.

Let $\displaystyle f(x)=\dfrac{1}{3x-5}$ and $g(x)=x^3-2\text{.}$

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(a)

Find $f(g(x))\text{.}$

$\displaystyle \dfrac{x^3-2}{3x-5}$

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$\displaystyle \dfrac{1}{(3x-5)(x^3-2)}$

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$\displaystyle \dfrac{1}{3(x^3-2)-5}$

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$\displaystyle \left( \dfrac{1}{3x-5} \right)^{3} -2$

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Answer.

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(b)

Find $g(f(x))\text{.}$

$\displaystyle \dfrac{x^3-2}{3x-5}$

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$\displaystyle \dfrac{1}{(3x-5)(x^3-2)}$

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$\displaystyle \dfrac{1}{3(x^3-2)-5}$

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$\displaystyle \left( \dfrac{1}{3x-5} \right)^{3} -2$

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Answer.

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Remark 2.5.8.

We can also evaluate the composition of two functions at a particular value just as we did with one function. For example, we may be asked to find something like $f(g(2))$ or $(g\circ f)(-3)\text{.}$

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Activity 2.5.9.

Let $\displaystyle f(x)=2x^3$ and $g(x)=\sqrt{6-x}\text{.}$

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(a)

Find $f(g(2))\text{.}$

$\displaystyle 14$

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$\displaystyle 16$

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$\displaystyle 18$

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$\displaystyle 20$

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undefined

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Answer.

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(b)

Find $(g \circ f)(-3)\text{.}$

$\displaystyle 50$

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$\displaystyle 54$

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$\displaystyle \sqrt{60}$

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$\displaystyle \sqrt{-48}$

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undefined

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Answer.

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(c)

Find $(f \circ g)(10)\text{.}$

$\displaystyle 2(\sqrt{-4})^3$

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$\displaystyle 16$

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$\displaystyle \sqrt{-1994}$

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$\displaystyle -16$

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undefined

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Answer.

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Remark 2.5.10.

As we saw in Activity 2.5.9, in order for a composite function to make sense, we need to ensure that the range of the inner function lies within the domain of the outer function so that the resulting composite function is defined at every possible input.

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Remark 2.5.11.

In addition to the possibility that functions are given by formulas, functions can be given by tables or graphs. We can think about composite functions in these settings as well, and the following activities prompt us to consider functions given in this way.

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Activity 2.5.12.

Let functions $p$ and $q$ be given by the graphs below.

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Find each of the following. If something is not defined, explain why.

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(a)

$(p \circ q)(0)$

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Answer.

$1$

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(b)

$q(p(0)) $

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Answer.

$2$

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(c)

$p(p(1))$

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Answer.

$-\dfrac{1}{2}$

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(d)

$(q \circ p)(-3)$

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Answer.

$0$

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(e)

Find two values of $x$ such that $q(p(x)) = 2\text{.}$

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Answer.

$x$-values on the interval $[-2,1.5]$

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Activity 2.5.13.

Let functions $f$ and $g$ be given by the tables below.

$x$	$f(x)$
$0$	$6$
$1$	$4$
$2$	$3$
$3$	$4$
$4$	$7$

Table 2.5.14.

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$x$	$g(x)$
$0$	$1$
$1$	$3$
$2$	$0$
$3$	$5$
$4$	$2$

Table 2.5.15.

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Find each of the following. If something is not defined, explain why.

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(a)

$(f \circ g)(2)$

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Answer.

$6$

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(b)

$(g \circ f)(3)$

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Answer.

$2$

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(c)

$g(f(4))$

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Answer.

Not defined because $f(4)=7$ and $7$ isn’t in the domain of $g(x)\text{.}$

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(d)

For what value(s) of $x$ is $f(g(x)) = 4\text{?}$

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Answer.

$x=0,1$

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(e)

What are the domain and range of $(f\circ g)(x)\text{?}$

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Answer.

Domain: $\{0,1,2,3,4 \}$ and Range: $\{3,4,6 \}$

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Remark 2.5.16.

Now that we’ve seen how to compose functions, let’s take a look at how composition of functions can be applicable in our lives.

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Activity 2.5.17.

Suppose you have a $\$50$-off coupon and there is a $15\%$-off sale on a TV. If you are allowed to apply both the coupon and sale price of the TV, which one should you apply first? Let’s investigate to determine the better deal.

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(a)

Write a cost function, $C(p)\text{,}$ for the price of the TV, $p\text{,}$ if you applied the $\$50$-off coupon.

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Answer.

$C(p)=p-50$

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(b)

Write a cost function, $S(p)\text{,}$ for the price of the TV, $p\text{,}$ if you applied the $15\%$-off sale.

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Answer.

$S(p)=p-.15p$

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$S(p)=.85p$

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(c)

Suppose you apply the coupon first and then the $15\%$-off sale. Write a composite function to represent this situation.

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Answer.

$S(C(p))=S(p-50)$

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$S(C(p))=.85(p-50)$

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$S(C(p))=.85p-42.5$

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(d)

Suppose you apply the $15\%$-off sale first and then the coupon. Write a composite function to represent this situation.

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Answer.

$C(S(p))=C(.85p)$

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$S(C(p))=(.85p)-50$

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(e)

Suppose the original cost of the TV is $\$500\text{.}$ Determine the cost of the TV if you applied the coupon first.

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Answer.

From part (c), we know that: $S(C(p))=.85p-42.5$

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$S(C(500))=.85(500)-42.5$

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$S(C(500))=382.50$

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When applying the coupon first, the reduced cost of the TV is $\$382.50\text{.}$

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(f)

Suppose the original cost of the TV is $\$500\text{.}$ Determine the cost of the TV if you applied the $15\%$-off sale first.

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Answer.

From part (d), we know that: $S(C(p))=(.85p)-50$

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$S(C(500))=(.85(500))-50$

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$S(C(500))=375$

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When applying the $15\%$-off sale first, the reduced cost of the TV is $\$375\text{.}$

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(g)

Which is the better deal and how much would you save?

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Answer.

When applying the coupon first, the TV would cost $\$382.50\text{.}$ If applying the $15\%$-off sale first, however, the TV would cost $\$375\text{.}$ Applying the $15\%$-off sale first is the better deal, saving you $\$125$ (versus $\$117.50$)!

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Subsection 2.5.2 Exercises

Exercises available at https://library.tbil.org/2025/precalculus/exercises/#/bank/FN5/

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\(x\)	\(f(x)\)
\(0\)	\(6\)
\(1\)	\(4\)
\(2\)	\(3\)
\(3\)	\(4\)
\(4\)	\(7\)

\(x\)	\(g(x)\)
\(0\)	\(1\)
\(1\)	\(3\)
\(2\)	\(0\)
\(3\)	\(5\)
\(4\)	\(2\)