1. 6.4 Use Inverse Functions

2. Recall from Chapter 2 x y -2 4 y = x2 -1 1 0 0 1 1
Relation – a mapping of input values (x-values) onto output values (y-values).
Here are 3 ways to represent the same relation.
x y
y = x2
Analytically
Numerically
Graphically

3. x y x = y2 -2 -1 0 0 1 1 Inverse relation – swap the x & y-values.
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. Find an inverse of y = -3x+6.
Steps: -swap x & y
-solve for y
y = -3x+6
x = -3y+6
x-6 = -3y
We could also write:

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. Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.
We must 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:
Change the f(x) to a y.
Switch the x & y values.
Solve the new equation for y.
** Remember functions have to pass the vertical line test!

8. Find the inverse of f(x)=x5.
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. 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)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.
y = 2x2-4
x = 2y2-4
x+4 = 2y2
f -1(x) is not a function.

12. Given: g(x)=2x3
y=2x3
x=2y3
Inverse is a function!