1.12. Inverses of Functions http://www.ck12.org
When given a function there are two steps to take to find its inverse. In the original function, first switch the
variablesxandy. Next, solve the function fory. This will give you the inverse function. After finding the inverse, it
is important to check both directions of compositions to make sure that together the function and its inverse produce
the valuex. In other words, verify thatf(f−^1 (x)) =xandf−^1 (f(x)) =x.
Graphically, inverses are reflections across the liney=x. Below you see inversesy=exandy=lnx. Notice how
the(x,y)coordinates in one graph become(y,x)coordinates in the other graph.
In order to decide whether an inverse function is also actually a function you can use the vertical line test on the
inverse function like usual. You can also use the horizontal line test on the original function. The horizontal line test
is exactly like the vertical line test except the lines simply travel horizontally.
Example A
Find the inverse, then verify the inverse algebraically.f(x) =y= (x+ 1 )^2 + 4
Solution:To find the inverse, switchxandythen solve fory.
x= (y+ 1 )^2 + 4
x− 4 = (y+ 1 )^2
±√x− 4 =y+ 1
− 1 ±√x− 4 =y=f−^1 (x)To verify algebraically, you must showx=f(f−^1 (x)) =f−^1 (f(x)):
f(f−^1 (x)) =f(
− 1 ±√x− 4)
= ((− 1 ±√x− 4 )+ 1 )^2 + 4
=(
±√x− 4) 2
+ 4
=x− 4 + 4 =x