3.4 Composition of Functions

1. ⓐ Evaluate $g \left(3\right)$
2. ⓑ Solve $g \left(n\right) = 6$

Find and simplify the functions $\left(g - f\right) \left(x\right)$ and $\left(\frac{g}{f}\right) \left(x\right) ,$ given $f \left(x\right) = x - 1$ and $g \left(x\right) = x^{2} - 1.$ Are they the same function?

Find and simplify the functions $\left(f g\right) \left(x\right)$ and $\left(f - g\right) \left(x\right) .$

Are they the same function?

Using the functions provided, find $f \left(g \left(x\right)\right)$ and $g \left(f \left(x\right)\right) .$ Determine whether the composition of the functions is commutative.

The function $c \left(s\right)$ gives the number of calories burned completing $s$ sit-ups, and $s \left(t\right)$ gives the number of sit-ups a person can complete in $t$ minutes. Interpret $c \left(s \left(3\right)\right) .$

Suppose $f \left(x\right)$ gives miles that can be driven in $x$ hours and $g \left(y\right)$ gives the gallons of gas used in driving $y$ miles. Which of these expressions is meaningful: $f \left(g \left(y\right)\right)$ or $g \left(f \left(x\right)\right) ?$

The gravitational force on a planet a distance r from the sun is given by the function $G \left(r\right) .$ The acceleration of a planet subjected to any force $F$ is given by the function $a \left(F\right) .$ Form a meaningful composition of these two functions, and explain what it means.

Using the table, evaluate $f \left(g \left(3\right)\right)$ and $g \left(f \left(3\right)\right) .$

Using the table, evaluate $f \left(g \left(1\right)\right)$ and $g \left(f \left(4\right)\right) .$

Using the figure, evaluate $g \left(f \left(2\right)\right) .$

Given $f \left(t\right) = t^{2} - t$ and $h \left(x\right) = 3 x + 2 ,$ evaluate $f \left(h \left(1\right)\right) .$

Given $f \left(t\right) = t^{2} - t$ and $h \left(x\right) = 3 x + 2 ,$ evaluate

1. ⓐ $h \left(f \left(2\right)\right)$
2. ⓑ $h \left(f \left(- 2\right)\right)$

Find the domain of

Find the domain of

Find the domain of

Write $f \left(x\right) = \sqrt{5 - x^{2}}$ as the composition of two functions.

Write $f \left(x\right) = \frac{4}{3 - \sqrt{4 + x^{2}}}$ as the composition of two functions.

How does one find the domain of the quotient of two functions, $\frac{f}{g} ?$

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

Determine the domain for each function in interval notation.

Given $f \left(x\right) = x^{2} + 2 x$ and $g \left(x\right) = 6 - x^{2} ,$ find $f + g , f - g , f g ,$ and $\frac{f}{g} .$

Determine the domain for each function in interval notation.

Given $f \left(x\right) = 2 x^{2} + 4 x$ and $g \left(x\right) = \frac{1}{2 x} ,$ find $f + g , f - g , f g ,$ and $\frac{f}{g} .$

Determine the domain for each function in interval notation.

Given $f \left(x\right) = 3 x^{2}$ and $g \left(x\right) = \sqrt{x - 5} ,$ find $f + g , f - g , f g ,$ and $\frac{f}{g} .$

Determine the domain for each function in interval notation.

For the following exercise, find the indicated function given $f \left(x\right) = 2 x^{2} + 1$ and $g \left(x\right) = 3 x - 5 .$

1. ⓐ $f \left(g \left(2\right)\right)$
2. ⓑ $f \left(g \left(x\right)\right)$
3. ⓒ $g \left(f \left(x\right)\right)$
4. ⓓ $\left(g \circ g\right) \left(x\right)$
5. ⓔ $\left(f \circ f\right) \left(- 2\right)$

Use each pair of functions to find $f \left(g \left(x\right)\right)$ and $g \left(f \left(x\right)\right) .$ Simplify your answers.

$f \left(x\right) = \sqrt{x} + 2 , g \left(x\right) = x^{2} + 3$

Use each pair of functions to find $f \left(g \left(x\right)\right)$ and $g \left(f \left(x\right)\right) .$ Simplify your answers.

$f \left(x\right) = \sqrt[3]{x} , g \left(x\right) = \frac{x + 1}{x^{3}}$

Use each pair of functions to find $f \left(g \left(x\right)\right)$ and $g \left(f \left(x\right)\right) .$ Simplify your answers.

$f \left(x\right) = \frac{1}{x - 4} , g \left(x\right) = \frac{2}{x} + 4$

Use each set of functions to find $f \left(g \left(h \left(x\right)\right)\right) .$ Simplify your answers.

$f \left(x\right) = x^{2} + 1 ,$ $g \left(x\right) = \frac{1}{x} ,$ and $h \left(x\right) = x + 3$

Use each set of functions to find $f \left(g \left(h \left(x\right)\right)\right) .$ Simplify your answers.

Given $f \left(x\right) = \sqrt{2 - 4 x}$ and $g \left(x\right) = - \frac{3}{x} ,$ find the following:

1. ⓐ $\left(g \circ f\right) \left(x\right)$
2. ⓑ the domain of $\left(g \circ f\right) \left(x\right)$ in interval notation

Use each set of functions to find $f \left(g \left(h \left(x\right)\right)\right) .$ Simplify your answers.

Given functions $p \left(x\right) = \frac{1}{\sqrt{x}}$ and $m \left(x\right) = x^{2} - 4 ,$ state the domain of each of the following functions using interval notation:

1. ⓐ $\frac{p \left(x\right)}{m \left(x\right)}$
2. ⓑ $p \left(m \left(x\right)\right)$
3. ⓒ $m \left(p \left(x\right)\right)$

Use each set of functions to find $f \left(g \left(h \left(x\right)\right)\right) .$ Simplify your answers.

For $f \left(x\right) = \frac{1}{x}$ and $g \left(x\right) = \sqrt{x - 1} ,$ write the domain of $\left(f \circ g\right) \left(x\right)$ in interval notation.

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \left(x - 5\right)^{3}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \frac{4}{\left(x + 2\right)^{2}}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \sqrt[3]{\frac{1}{2 x - 3}}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \sqrt[4]{\frac{3 x - 2}{x + 5}}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \sqrt{2 x + 6}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \sqrt[3]{x - 1}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \frac{1}{\left(x - 2\right)^{3}}$

Find functions $f \left(x\right)$ and $g \left(x\right)$ so the given function can be expressed as $h \left(x\right) = f \left(g \left(x\right)\right) .$

$h \left(x\right) = \sqrt{\frac{2 x - 1}{3 x + 4}}$

$f \left(g \left(1\right)\right)$

$g \left(f \left(0\right)\right)$

$f \left(f \left(4\right)\right)$

$g \left(g \left(0\right)\right)$

$g \left(f \left(2\right)\right)$

$f \left(g \left(1\right)\right)$

$h \left(f \left(2\right)\right)$

$f \left(g \left(f \left(- 2\right)\right)\right)$

$f \left(g \left(5\right)\right)$

$g \left(f \left(3\right)\right)$

$f \left(f \left(1\right)\right)$

$g \left(g \left(6\right)\right)$

$\left(f \circ g\right) \left(2\right)$

$\left(g \circ f\right) \left(3\right)$

$\left(f \circ f\right) \left(3\right)$

Use each pair of functions to find $f \left(g \left(0\right)\right)$ and $g \left(f \left(0\right)\right) .$

$f \left(x\right) = 5 x + 7 , g \left(x\right) = 4 - 2 x^{2}$

Use each pair of functions to find $f \left(g \left(0\right)\right)$ and $g \left(f \left(0\right)\right) .$

$f \left(x\right) = \frac{1}{x + 2} , g \left(x\right) = 4 x + 3$

Use the functions $f \left(x\right) = 2 x^{2} + 1$ and $g \left(x\right) = 3 x + 5$ to evaluate or find the composite function as indicated.

$f \left(g \left(x\right)\right)$

Use the functions $f \left(x\right) = 2 x^{2} + 1$ and $g \left(x\right) = 3 x + 5$ to evaluate or find the composite function as indicated.

$\left(g \circ g\right) \left(x\right)$

Use $f \left(x\right) = x^{3} + 1$ and $g \left(x\right) = \sqrt[3]{x - 1} .$

Find $\left(f \circ g\right) \left(2\right)$ and $\left(g \circ f\right) \left(2\right) .$

Use $f \left(x\right) = x^{3} + 1$ and $g \left(x\right) = \sqrt[3]{x - 1} .$

What is the domain of $\left(f \circ g\right) \left(x\right) ?$

Let $F \left(x\right) = \left(x + 1\right)^{5} ,$ $f \left(x\right) = x^{5} ,$ and $g \left(x\right) = x + 1.$

True or False: $\left(g \circ f\right) \left(x\right) = F \left(x\right) .$

Find the composition when $f \left(x\right) = x^{2} + 2$ for all $x \geq 0$ and $g \left(x\right) = \sqrt{x - 2} .$

$\left(f \circ g\right) \left(6\right) ; \left(g \circ f\right) \left(6\right)$

Find the composition when $f \left(x\right) = x^{2} + 2$ for all $x \geq 0$ and $g \left(x\right) = \sqrt{x - 2} .$

$\left(f \circ g\right) \left(11\right) ; \left(g \circ f\right) \left(11\right)$

The function $A \left(d\right)$ gives the pain level on a scale of 0 to 10 experienced by a patient with $d$ milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after $t$ minutes is modeled by $m \left(t\right) .$ Which of the following would you do in order to determine when the patient will be at a pain level of 4?

1. ⓐEvaluate $A \left(m \left(4\right)\right) .$
2. ⓑEvaluate $m \left(A \left(4\right)\right) .$
3. ⓒSolve $A \left(m \left(t\right)\right) = 4.$
4. ⓓSolve $m \left(A \left(d\right)\right) = 4.$

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to $r \left(t\right) = 25 \sqrt{t + 2} ,$ find the area of the ripple as a function of time. Find the area of the ripple at $t = 2.$

Use the function you found in the previous exercise to find the total area burned after 5 minutes.

The number of bacteria in a refrigerated food product is given by $N \left(T\right) = 23 T^{2} - 56 T + 1 ,$ $3 < T < 33 ,$ where $T$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $T \left(t\right) = 5 t + 1.5 ,$ where $t$ is the time in hours.

1. ⓐFind the composite function $N \left(T \left(t\right)\right) .$
2. ⓑFind the time (round to two decimal places) when the bacteria count reaches 6752.