1. 3
Inverse Functions

2. Relation – a mapping of input values (x-values) onto output values (y-values).
Here are 3 ways to show the same relation.
x y
y = x2
Equation
Table of values
Graph

3. Inverse relation – just think: switch the x & y-values.-2 -1
x = y2
** the inverse of an equation: switch the x & y and solve for y.
** the inverse of a table: switch the x & y.
** the inverse of a graph: the reflection of the original graph in the line y = x.

4. Ex: Find an inverse of y = -3x+6.
Steps: -switch x & y
-solve for y
y = -3x+6
x = -3y+6
x-6 = -3y

5. Inverse Functions
Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other.
Symbols: f -1(x) means “f inverse of x”

6. Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.
Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.
f(g(x))= -3(-1/3x+2)+6
= x-6+6
= x
g(f(x))= -1/3(-3x+6)+2
= x-2+2
= x
** Because f(g(x))=x and g(f(x))=x, they are inverses.

7. To find the inverse of a function:
Switch the x & y values.
Solve the new equation for y.
** Remember functions have to pass the vertical line test!

8. Ex: (a)Find the inverse of f(x)=x5.
(b) Is f -1(x) a function?
(hint: look at the graph!
Does it pass the vertical line test?)
y = x5
x = y5
Yes , f -1(x) is a function.

9. Horizontal Line Test
Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test.
If the original function passes the horizontal line test, then its inverse is a function.
If the original function does not pass the horizontal line test, then its inverse is not a function.

10. Ex: Graph the function f(x)=x2 and determine whether its inverse is a function.
Graph does not pass the horizontal line test, therefore the inverse is not a function.

11. Ex: f(x)=7x-1 Determine whether f -1(x) is a function, then find the inverse equation.
y = 7x - 1
x = 7y- 1
x+ 1 = 7y
f -1(x) is a function.