1. 3.4 Use Inverse Functions p. 190 What is an inverse relation?
What do you switch to find an inverse relation?
What notation is used for an inverse function?
How is it read?
What test can you use to verify the inverse of a function is a function?

2. Review
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. Find an equation for the inverse of the relation
y = 3x – 5.
y = 3x – 5
Write original relation.
x = 3y – 5
Switch x and y.
x + 5 = 3y
Add 5 to each side. 1 3
x + 5
= y
Solve for y. This is the inverse relation.

6. 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”

7. 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.

8. 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.

9. SOLUTION STEP 1 STEP 2 Show: that f(f –1(x)) = x.
Verify that f(x) = 3x – 5 and f –1(x) = 1 3 x + 5
are inverse functions.
SOLUTION
STEP 1
STEP 2
Show: that f(f –1(x)) = x.
Show: that f –1(f(x)) = x.
f (f –1(x)) = f 3 1
x + 5
f –1(f(x)) =
f –1(3x – 5) 3 1
x + 5
= 3
– 5 = 1 3 5
(3x – 5) +
= x – 5 3 +
= x + 5 – 5
= x
= x

10. Fitness 3 8
Elastic bands can be used in exercising to provide a range of resistance. A band’s resistance R (in pounds) can be modeled by R = L – 5 where L is the total
length of the stretched band (in inches).

11. •
Find the inverse of the model.
Use the inverse function to find the length at which the band provides 19 pounds of resistance. •
SOLUTION
STEP 1
Find: the inverse function.
R =
L – 5 3 8
Write original model.
R + 5 = 3 8 L
Add 5 to each side. 8 3 40
R +
= L
Multiply each side by 8 3 .

12. Evaluate: the inverse function when R = 19.40 3
L = 8
R + 8 3 =
(19) + 40 40 3
152 = +
192 3 =
= 64
ANSWER
The band provides 19 pounds of resistance when it is stretched to 64 inches.

13. Find the inverse of f(x) = x2, x ≥ 0. Then graph f and f –1.
SOLUTION
f(x) = x2
Write original function.
y = x2
Replace f (x) with y.
x = y2
Switch x and y. = x y
Take square roots of each side.
The domain of f is restricted to nonnegative values of x. So, the range of f –1 must also be restricted to nonnegative values, and therefore the inverse is f –1(x) = x. (If the domain was restricted to x ≤ 0, you would choose f –1(x) = – x.)

14. Vertical Line Test
A vertical line test for functions can be used to see if the relation is a function.
A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.

15. 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.

16. 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.

17. Ex: g(x)=2x3 y=2x3 x=2y3 Inverse is a function!
OR, if you fix the tent in the basement…
Inverse is a function!

18. Consider the function f (x) = 2x3 + 1
Consider the function f (x) = 2x Determine whether the inverse of f is a function. Then find the inverse.
SOLUTION
Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is itself a function. To find an equation for f –1, complete the following steps:

19. Find the inverse of a cubic funtion
f (x) = 2x3 + 1
Write original function.
y = 2x3 + 1
Replace f (x) with y.
x = 2y3 + 1
Switch x and y.
x – 1 = 2y3
Subtract 1 from each side.
x – 1 2
= y3
Divide each side by 2. 3
x – 1 2
= y
Take cube root of each side.
The inverse of f is f –1(x) = 3
x – 1 2 .

20. Find the inverse of the function
Find the inverse of the function. Then graph the function and its inverse.
5. f(x) = x6, x ≥ 0
ANSWER
f –1(x) = 6√ x

21. 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.

22. 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
OR, if you fix the tent in the basement…
f -1(x) is not a function.

23. What is an inverse relation?
It maps the output values back to the original input values (domain or inverse is range or original).
What do you switch to find an inverse relation?
x and y
What notation is used for an inverse function?
f -1 (x)
How is it read?
f inverse of x
What test can you use to verify the inverse of a function is a function?
Horizontal line test

24. Assignment
Page194, 3-39 every 3rd problem, 46, 47