Advanced Algebra with Trigonometry

4.1 Inverse Functions

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

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)}

Was the original relation a function? What about the inverse?

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.

ff−1 f−1fx → → f(x) → → x x → → f−1(x) → → x

Example] Let f(x) = x + 2. f -1(x) = x – 2 Let x = 5.

5 → 5 + 2 → 7 7 → 7 – 2 → 5 5 → 5 – 2 → 3 3 → 3 + 2 → 5

{(3, 2), (6, 5), (3, 9)

Yes No . . . 3 repeats

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 + 1 (linear) Think: multiply by 3; then add 1

Write as y =; then switch x and y.y = 3x + 1

x = 3y + 1 Now, solve for y.

3y + 1 = x This step just makes it easier on the eye.

3y = x – 1

31x

y

Think: subtract 1; then divide by 3.

1 1( )

3

xf x

Rename y as f –1(x)

Check both function compositions to insure we found the correct inverse!!

f(f –1(x)) = f = 3 + 1

31x

31x

= x – 1 + 1

= x

(f –1(f(x)) = f –1(3x + 1) =3

11x3

= 3x3

= x

f(x) = 3x + 11 1( )

3

xf x

f(x) = x2 + 2, x ≥ 0 (Note the domain restriction: we are only interested in the right side of the parabola.)

f−1(x) =

y = x2 + 2

x = y2 + 2Switch x and y.

y2 + 2 = x Easier on the eye.

y2 = x − 2

Solve for y.

y = 2x

2x Because the original domain was restricted to non-zero reals, you are only interested in the positive portion of the solution.

Check both function compositions to insure we found the correct inverse!!

f(f –1(x)) = f = 2x 22x2

= x – 2 + 2

= x

(f –1(f(x)) = f –1(x2 + 2) = 22x2

= 2x

= x

f(x) = x2 + 2, x ≥ 0 2x f−1(x) =

Try this example: f(x) 3x2

1

y 3x2

1

x 3y21

x3y21

3xy21

)3x(2y

Quick check:

Let x = 8 f(x) = 7

f –1(7) = 8

1( ) 2( 3)f x x

Try this example: f(x) = x3 + 5

y = x3 + 5

x = y3 + 5

y3 = x – 5

3 5xy

y3 + 5 = x

Quick check:

f(2) = 13

f –1 (13) = 2

31 5 xf

Is the graph a function??? Use the Vertical Line Test!!!

Is the graph’s inverse a function??? Use the Horizontal Line Test!!!

OOPS!!

The graph’s inverse is NOT a function.

Remember: The graphs of inverses are reflections of each other in the line y = x.

f(x)= x2

f -1(x)

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:

Then I would plot (−1, −2), (0, 0), and (3, 2).Now it is easier to sketch the inverse!!

I would write down the coordinates of the three points indicated:(−2, −1), (0, 0), and (2, 3).

f(x)

f -1(x)

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