1.2.4What is a composite function?

Composite and Inverse Functions

1-61.

Given the two functions $f$ and $g$ graphed below:

State domain and range of $f$ .
State domain and range of $g$ .
What is $f(g(–2))$ ?
What is $g(f(–2))$ ?
What is $f(f(3))$ ?
Why is $f(g(5))$ undefined?

Continuous linear Piecewise graph, labeled, f of x, left segment, from (negative 2, comma 3), to (2, comma negative 1), right segment from (2, comma negative 1), to (4, comma 1).

Decreasing curve labeled, g of x, starting at (negative 6, comma 3), changing from opening down to opening up at (negative 1, comma 0), passing through approximate point (0, comma negative 2), ending at (6, comma negative 3).

1-62.

If $f(x) = x^2$ , $g(x) = x + 1$ , and $h(x) =\frac { 1 } { x }$ , express each given function as a composition of $f$ , $g$ , and/or $h$ .

For example, $k(x) = (x + 1)^2$ can be expressed as $k(x) = f(g(x))$ .

$j(x) =\frac { 1 } { x ^ { 2 } }$
$m(x) =\frac { 1 } { x }+1$
$p(x) = x^4$
$n(x) =\frac { 1 } { x ^ { 2 } + 1 }$

Explore this using the 1-62 Student eTool. Click in the lower right corner of the graph to view it in full-screen mode.

1-63.

Given $f(x) = 2^x$ and $g(x) =\sqrt { 1 - x }$ :

State the domains and ranges of $f$ and $g$ . Use interval notation.
Write an equation for $f(g(x))$ and state its domain.
Write an equation for $g(f(x))$ and state its domain.

1-64.

INVERSE FUNCTIONS

Let $h(x) = 3x + 2$ and $j(x) =\frac { x - 2 } { 3 }$ .

Write an equation for $h(j(x))$ . What do you notice?
Functions such that $f(g(x)) = g(f(x)) = x$ are called inverse functions. Explain why this notation shows that $f$ and $g$ are inverse functions.
If $f(x) = e^x + 2$ , write the equation of a function $g$ such that $f(g(x)) = x$ .

1-65.

An inverse function undoes what a function does. For example, $\sin(\ \frac{\pi}{6})=\frac{1}{2}$ , which means the sine function takes the angle $\frac { \pi } { 6 }$ and returns the ratio $\frac { 1 } { 2 }$ . Therefore the inverse sine function takes the ratio $\frac { 1 } { 2 }$ and returns the angle $\frac { \pi } { 6 }$ . The notation for inverse functions can be confusing; the inverse of $f$ is written $f^{ -1}$ . The inverse sine function is written $\sin^{–1}(x)$ . $\sin^{–1}(x)$ is also referred to as $\arcsin(x)$ . Note: $\sin^{-1}(x)\ne\frac{1}{\sin(x)}$ !

Write each of the statements below entirely in symbols.

The inverse sine of $\frac { 1 } { 2 }$ is $\frac { \pi } { 6 }$ .

When the inverse of the function $g$ is applied to $7$ , the result is $5$ .

Inverse Functions

Upward parabola labeled, y = x squared + 2, vertex at (0, comma 2), left half is dashed, dashed increasing line labeled, y = x, going through the origin, increasing curve, labeled, y = square root of the quantity, x minus 2), starting at (2, comma 0), increasing up & right. We say that $f$ and $g$ are inverse functions if $f\left(g\left(x\right)\right) = x$ for all $x$ in the domain of $g$ and $g\left(f\left(x\right)\right) = x$ for all $x$ in the domain of $f$ . We write $f^{ –1}$ for the inverse of $f$ . So $g\left(x\right) = f^{ –1}\left(x\right)$ and $f \left(x\right) = g^{–1}\left(x\right)$ .

Note: The domains of some functions must be restricted in order for the inverses to be inverse functions of each other.

For example $f\left(x\right) = x^{2} + 2$ for $x\ge0$ and $g(x)=\sqrt{x-2}$ for $x\ge2$ .

If we graph $y = f\left(x\right)$ and $y = f^{ –1}\left(x\right)$ on the same set of axes then their graphs are symmetric across the line $y = x$ .

If a function satisfies the horizontal line test, then an inverse function exists.

1-66.

In parts (a) and (b), solve for $x$ .

$f(x) = 2^x$

$g(x) =\frac { x + 1 } { x }$

Now write the inverse equations of $f$ and $g$ . What do you notice?

1-67.

Study the table for the functions $f$ and $g$ at right. $f$ does not have an inverse function. Explain why not.
Evaluate:
1. $g^{-1}(2)$
1. $f(g^{-1}(2))$
1. $g^{-1}(g(-2))$
If $h(3) = 4$ and $j(x) = h^{–1}(x)$ , what is $j(4)$ ?

$x$	$f(x)$	$g(x)$
$–2$	$5$	$–3$
$–1$	$8$	$–1$
$0$	$9$	$0$
$1$	$8$	$2$
$2$	$5$	$3$

1-68.

Write a possible equation for each of the following graphs. Verify your equations on your graphing calculator. 1-69 HW eTool. Homework Help ✎

1-69.

Much of this course will focus on examining how functions grow. Examine two ways a straight line grows by completing the parts below. 1-69 HW eTool. Homework Help ✎

Sketch $f(x) = 2x + 3$ . What are $f\left(0\right)$ , $f\left(1\right)$ , $f\left(2\right)$ , and $f\left(3\right)$ ? How does $f$ grow as $x$ increases?
Sketch $g(x)=-3x+10$ . What are $g\left(0\right)$ , $g\left(1\right)$ , $g\left(2\right)$ , and $g\left(3\right)$ ? How does $g$ grow as $x$ increases?

1-70.

Selected values of a continuous even function are shown below. 1-70 HW eTool Homework Help ✎

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

What are $f\left(-1\right)$ , $f\left(-2\right)$ , and $f\left(-3\right)$ ?
Sketch a possible graph of the function on the domain $–3 ≤ x ≤ 3$ .
Sketch another possible graph of the function on the domain $−3 ≤ x ≤ 3$ .
Can the graph be a quadratic function? If so, write a possible equation for the function. If not, explain why not.

1-71.

State the domain of each of the following functions. Homework Help ✎

$f(x) = \sqrt { x + 2 }$

$g(x) =\frac { 1 } { x - 4 }+3$

$h(x) = \log(x - 4)$

$j(x) =\sqrt { \frac { 2 - x } { x } }$

1-72.

Helen thinks $\sqrt { x ^ { 2 } }= x$ . Felicia does not agree. Homework Help ✎

Use various values of $x$ to check whether or not Helen is correct.
Write an accurate expression for $\sqrt { x ^ { 2 } }$ .

1-73.

Sketch $f(x) = 3\sqrt { x + 1 }$ on $0 ≤ x ≤ 6$ three times, on three different sets of axes. 1-73 HW eTool Homework Help ✎

Review your work from problems 1-25 and 1-36. Use a similar process to approximate the area under the curve for $0 ≤ x ≤ 6$ using:
1. Six left endpoint rectangles.
2. Six right endpoint rectangles.
3. Six trapezoids.
Which approximations were overestimates of (greater than) the actual area? Which were underestimates? Explain.
Which approximation is the most accurate? Explain.

1-74.

Use interval notation to state the domain and range of each function below. Homework Help ✎

1-75.

A can of soda is $42^\circ\text{ F}$ when purchased. Over the course of the next few hours, the temperature of the soda slowly rises. During an experiment, Shibisha used a thermometer and recorded the temperature at various times, $t$ , shown in the table below. 1-75 HW eTool Homework Help ✎

Time (min)	$0$	$10$	$30$	$45$	$60$	$75$	$80$
Temp. ( $\bf^\circ \text{F}$ )	$42$	$51$	$58$	$63$	$66$	$68$	$69$

Sketch a graph of this situation.
When is the temperature changing the fastest? How can you see this on the graph?
Approximately how fast is the temperature changing during the first $10$ minutes? How can you tell?

1-76.

Each of the continuous functions in the table below is increasing, but each increases differently. Match each graph below with the function that grows in a similar fashion in the table. Homework Help ✎

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$f (x)$	$15.5$	$19.0$	$22.5$	$26.0$	$29.5$	$33.0$	$36.5$	$40.0$	$43.5$
$g(x)$	$1$	$2$	$4$	$8$	$16$	$32$	$64$	$128$	$256$
$h(x)$	$12$	$76$	$108$	$124$	$132$	$136$	$138$	$139$	$139.5$

Calculus Third Edition

Inverse Functions