1. Advanced Algebra with Trigonometry
4.1 Inverse Functions

2. Inverse Concepts Stand Up Sit Down Go To Sleep Wake Up
2 Steps Forward, then turn Right
Turn Left, then 2 Steps Backward
Square, Mulitply, then Add
Subtract, Divide, then Square Root

3. {(3, 2), (6, 5), (3, 9) Yes No . . . 3 repeats x → f f−1 x → f−1
4.1 Inverse Functions
Inverse relation: a mapping of output values back to their original input values
In plain English: switch the x and y!! (x, y) becomes (y, x)!!
Write the inverse relation: {(2, 3), (5, 6), (9, 3)}
{(3, 2), (6, 5), (3, 9)
Was the original relation a function? What about the inverse?
Yes
No repeats
If both the original relation and the inverse relation are functions, we call the two functions inverse functions. NOTE: f −1 is read as f inverse.
Two functions f and f −1 are inverses of each other provided that
f (f −1(x)) = x and f −1(f (x)) = x.
Example] Let f(x) = x f -1(x) = x – Let x = 5.
x → f
→ f(x) →
f−1
→ x
x →
f−1
→ f−1(x) → f
→ x
5 → → → 7 – 2 → 5
5 → 5 – 2 → → → 5

4. f(x) = 3x + 1 (linear) y = 3x + 1 x = 3y + 1 3y + 1 = x 3y = x – 1
Steps to find inverse:
1. Switch x and y
2. Solve for y
Think of the inverse as “undoing” the original function.
f(x) = 3x (linear)
Think: multiply by 3; then add 1
y = 3x + 1
Write as y =; then switch x and y.
x = 3y + 1
Now, solve for y.
3y + 1 = x
This step just makes it easier on the eye.
3y = x – 1
Think: subtract 1; then divide by 3.
Rename y as f –1(x)

5. Check both function compositions to insure we found the correct inverse!!
f(x) = 3x + 1
f(f –1(x)) = f =
= x –
= x
(f –1(f(x)) = f –1(3x + 1) = =
= x

6. f(x) = x2 + 2, x ≥ 0. (Note the domain restriction: we are only
f(x) = x , x ≥ 0 (Note the domain restriction: we are only interested in the right side of the parabola.)
y = x
Switch x and y.
x = y
y = x
Easier on the eye.
Solve for y.
y2 = x − 2
y =
f−1(x) =
Because the original domain was restricted to non-zero reals, you are only interested in the positive portion of the solution.

7. Check both function compositions to insure we found the correct inverse!!
f(x) = x , x ≥ 0
f−1(x) =
f(f –1(x)) = f =
= x –
= x
(f –1(f(x)) = f –1(x ) = =
= x

8. Try this example: f(x) y
Quick check:
Let x = f(x) = 7
f –1(7) = 8 x

9. Try this example: f(x) = x3 + 5
Quick check:
f(2) = 13
f –1 (13) = 2
y = x3 + 5
x = y3 + 5
y3 + 5 = x
y3 = x – 5

10. Is the graph a function??? Use the Vertical Line Test!!!
Is the graph’s inverse a function??? Use the Horizontal Line Test!!!
Remember: The graphs of inverses are reflections of each other in the line y = x.
f(x)= x2
OOPS!!
The graph’s inverse is NOT a function.
f -1(x)

11. Some people can just visualize what an inverse graph looks like using the definition. Others of us need a little assistance.
My secret:
On the original graph, pick two or three easily identified points.
Write them down.
Switch the x and y coordinates!!!
Plot the new points to give you a start on drawing the inverse.
Graph the line y = x.
The inverse graph will be symmetric about the line y = x.
Consider the function shown in the graph:
f(x)
I would write down the coordinates of the three points indicated:
(−2, −1), (0, 0), and (2, 3).
f -1(x)
Then I would plot (−1, −2), (0, 0), and (3, 2).
Now it is easier to sketch the inverse!!

12. Review: To find the inverse of a relation or function:
To test if two relations are functions:
Don’t forget about any domain restrictions!
Switch x and y.
Solve for y.
Use Composition to verify that
f(g(x)) equals x and g(f(x)) equals x