LN #11 (Notes #25)
Lesson 1.4 Inverse Functions
A. What is it?
1. Function- a relation where for every x value, there is only one y-value. (Use vertical line test to determine if a relation
is a function.)
2. Inverse Function- a function that undoes the action of another function.
3. Inverse of a function- the relation formed when the independent variable (x) is exchanged with the dependent
variable (y) in a given relation. (This inverse may or may not be a function.)
If the inverse is a function, the notation is used:
Original: f(x)
Inverse: f-1(x)
Algebraic Examples: (In these examples, “y” is the same as “f(x).”
Original Function
1. Exchange x for y
2. Rearrange to isolate y.
3. Rewrite y as f-1(x).
Inverse of the Function
Is the Inverse a Function?
Example 1
f(x) = 3x + 9
y = x + 9  x = 3y + 9
–9 –9
x – 9 = 3y
3
3
1
𝑥
−
3
=
𝑦
3
f-1(x) = 13𝑥 − 3
Example 2
f(x) = x2 + 1
y = x2 + 1  x = y2 + 1
–1
–1
√𝑥 − 1 = √𝑦 2
±√𝑥 − 1 = 𝑦
f (x) = ±√𝑥 − 1, 𝑥 ≥ 1
-1
Graphic Examples:
Example 1:
Example 2:
f(x) = 3x + 9
f(x) = x2 + 1
f-1(x) = 13𝑥 − 3
B. How do you know if the inverse is a function?
The original function must be a one-to-one function to guarantee that its inverse will also be a function.
f-1(x) = ±√𝑥 − 1
Use the horizontal line test. If the graph of the original function only crosses the horizontal line ONE TIME in each
location, then the function is a one-to-one function. Therefore, its inverse will also be a function.
~~~~~~~~~(From this point forward, only copy what is surrounded by a red box and highlighted in yellow.)~~~~~~~~~~
C. How to find inverses
There are 3 ways to find the inverse of a function.
1. Swap ordered pairs
2. Solve algebraically (As shown above in example 1 and 2).
3. Graph
For more information, see http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm