CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 1. Functions and Graphs

f−^1 (f(x)) =f−^1 ((x+ 1 )^2 + 4 )

=− 1 ±

√

((x+ 1 )^2 + 4 )− 4

=− 1 ±

√

(x+ 1 )^2

=− 1 +x+ 1 =x

As you can see from the graph, the±causes the inverse to be a relation instead of a function. This can be observed
in the graph because the original function does not pass the horizontal line test and the inverse does not pass the
vertical line test.

Example B
Find the inverse of the function and then verify thatx=f(f−^1 (x)) =f−^1 (f(x)).
f(x) =y=xx+−^11
Solution:Sometimes it is quite challenging to switchxandyand then solve fory. You must be careful with your
algebra.

x=yy+−^11

x(y− 1 ) =y+ 1

xy−x=y+ 1

xy−y=x+ 1

y(x− 1 ) =x+ 1

y=xx+−^11

This function turns out to be its own inverse. Since they are identical, you only need to show thatx=f(f−^1 (x)).

f(xx+−^11 )=(

xx+− (^11) )+ 1
(xx−+^11 )−^1 =
xx++ 11 −+(xx−−^11 )=^22 x=x
Example C