Chapter Fifteen — Functions
The Humongous Book of Algebra Problems
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.
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
“the inverse of
f(x).” It does NOT
mean “f(x) raised to
the –1 power.” That
would be written