Chapter Fifteen — Functions

The Humongous Book of Algebra Problems

335

Inverse Functions

Functions that cancel each other out

15.27 What is the deﬁning characteristic of inverse functions?

If functions f(x) and g(x) are inverse functions, then composing one function

with the other—and vice versa—results in x: f(g(x)) = g(f(x)) = x.

15.28 Describe the relationship between the points on the graph of a function and

the points on the graph of its inverse.

If the graph of function f(x) contains the point (a,b), then the graph of its

inverse function contains the point (b,a). Reversing the coordinates of

one graph produces the coordinates for the graph of its inverse.

15.29 Does an inverse function exist for the function f(x) graphed in Figure 15-1?

Why or why not?

Figure 15-1: No information about f(x) is provided other than this graph.

Function f(x) does not have an inverse function because it fails the horizontal

line test, which states that a function is one-to-one if and only if any horizontal

line drawn on its graph will, at most, intersect the graph only once. Notice that

any horizontal line drawn on Figure 15-1 between y = 1 and y = 3 will intersect

the graph three times.

In other

words, inverse

functions cancel

each other out.

Plugging g(x) into f(x) or

f(x) into g(x) makes f(x)

and g(x) go away,

leaving only x

behind.

“f

–1

(x)” means

“the inverse of

f(x).” It does NOT

mean “f(x) raised to

the –1 power.” That

would be written

[f(x)]

–1

.

Chapter Fifteen — Functions

The Humongous Book of Algebra Problems

336

Every input value of a function corresponds to exactly one output value, but the

output values of one-to-one functions correspond to exactly one input value as

well. If a horizontal line intersects a graph multiple times, the x-values of those

intersection points represent input values that are paired with an identical

output (the shared y-value represented by the horizontal line).

Consider the horizontal line y = h on the graph of f(x) in Figure 15-2. It

intersects the graph at three points, with x-values a, b, and c. Therefore, f(a) = h,

f(b) = h, and f(c) = h.

Figure 15-2: The horizontal line y = h intersects f(x) when x = a, x = b, and x = c.

As stated in Problem 15.28, the coordinate pairs of functions and their inverses

are reversed; therefore, , , and . However,

each input value of a function can correspond to only one output value, so the

relation is not a function.

RULE

OF THUMB:

If ANY horizontal

line intersects a

graph in more than

one place, the graph

fails the horizontal

line test. That

means the graph is

not one-to-one, and

only one-to-one

functions have

inverses.

Chapter Fifteen — Functions

The Humongous Book of Algebra Problems

337

15.30 Does an inverse function exist for the function g(x) graphed in Figure 15-3?

Why or why not?

Figure 15-3: No information about g(x) is provided other than this graph.

Function g(x) in Figure 15-3 passes the horizontal line test, as it is monoton-

ically increasing. According to the logic presented in Problem 15.29, g(x) is a

one-to-one function, and therefore the inverse function g

–1

(x) exists.

“Monotonic”

means “doesn’t

change direction.”

As you trace the

function’s graph from

left to right, it always

travels upward. At no

time does the graph

turn and start to

travel downwards

again.

Chapter Fifteen — Functions

The Humongous Book of Algebra Problems

338

15.31 Given the graph of function h(x) in Figure 15-4, draw the graph of the inverse

function, h

–1

(x).

Figure 15-4: The graph of h(x) passes through points (–4,3), (0,2), (3,0), (5,–2).

According to Problem 15.28, the graphs of a function and its inverse contain

reversed coordinate pair—if (a,b) belongs to the graph of h(x), then the graph

of h

–1

(x) contains the point (b,a). In this case, the graph of h

–1

(x) passes through

points (–2,5), (0,3), (2,0), and (3,–4). As a result the graphs are symmetric

about the line y = x, as illustrated by Figure 15-5.

Figure 15-5: The graphs of h

–1

(x) and h(x) are reﬂections of each other across the line

y = x.

These

are the

points that

h(x) passes

through, after

you reverse the

coordinates. For

example, h(x) passes

through (5,–2) so

h

–1

(x) passes

through

(–2,5).

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