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).
Chapter Fifteen — Functions
The Humongous Book of Algebra Problems
339
15.32 Verify that and are inverse functions.
As stated in Problem 15.27, if f(x) and g(x) are inverse functions, then
f(g(x)) = g(f(x)) = x. Substitute g(x) into f(x) (and vice versa) to verify that
both result in x.
Because f(g(x)) = g(f(x)) = x, f(x) and g(x) are inverse functions.
Note: Problems 15.33–15.34 refer to the function j(x) = 3(x + 4) – 1.
15.33 Identify the inverse function, j
–1
(x).
Begin by rewriting j(x) as y.
y = 3(x + 4) – 1
As Problem 15.28 states, the points on the graph of a function and the points
on the graph of its inverse contain reversed coordinate pairs. To reflect this
property of inverses, reverse x and y in the equation.
x = 3(y + 4) – 1
Solve the equation for y.
Replace y with j
–1
(x) to identify the inverse function.
If a fraction
contains a sum or
difference in the
numerator, you should
rewrite it as the sum
or difference of two
fractions (that have
the same denominator
as the big fraction
did), just like in Problem
15.26.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.