Inverse FunctionsInverse Functions

Given 2 functions, f(x) & g(x), if f(g(x))=x Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are AND g(f(x))=x, then f(x) & g(x) are inverses of each other.inverses of each other.

Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)=-3x+6 and g(x)=Ex: Verify that f(x)=-3x+6 and g(x)=-1-1//33x+2 are x+2 are

inverses.inverses. Meaning find f(g(x)) and g(f(x)). If they both Meaning find f(g(x)) and g(f(x)). If they both

equal x, then they are inverses.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, ** Because f(g(x))=x and g(f(x))=x, they are inversesthey are inverses..

Your Turn!Your Turn! Given that Given that ff((xx) = 7) = 7xx 2, use composition of 2, use composition of

functions to show that functions to show that ff 11((xx) = () = (xx + 2)/7. + 2)/7.

Solution:Solution: 1 1

1

( )( ) ( ( ))

(7 2)

(7 2) 2

77

7

f f x f f x

f x

x

x

x

1 1( )( ) ( ( ))

2( )72

7( ) 272 2

f f x f f x

xf

x

x

x

To find the inverse of a function:To find the inverse of a function:

1.1. Change the f(x) to a y.Change the f(x) to a y.

2.2. Switch the x & y values.Switch the x & y values.

3.3. Solve the new equation for y.Solve the new equation for y.

** Remember ** Remember functionsfunctions have to pass the have to pass the vertical line test!vertical line test!

To find the inverse of a function:To find the inverse of a function:

1.1. Change the f(x) to a y.Change the f(x) to a y.

2.2. Switch the x & y values.Switch the x & y values.

3.3. Solve the new equation for y.Solve the new equation for y.

** Remember ** Remember functionsfunctions have to pass the have to pass the vertical line test!vertical line test!

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

-solve for y-solve for y

y = -3x+6y = -3x+6

x = -3y+6x = -3y+6

x-6 = -3yx-6 = -3y

yx

3

6

23

1

xy

Ex: g(x)=2xEx: g(x)=2x33

Inverse is a function!

y=2x3

x=2y3

3

2y

x

yx

3

2

3

2

xy

Ex: f(x)=2xEx: f(x)=2x22-4 Determine whether f-4 Determine whether f -1 -1(x) is a (x) is a function, then find the inverse equation.function, then find the inverse equation.

2

2

4y

x

f -1(x) is not a function.

y = 2x2-4

x = 2y2-4

x+4 = 2y2

2

4x

y

22

1 xy

Your Turn!Your Turn!f(x) = 3xf(x) = 3x22 - 4 - 4

x=3y2 - 4

24 3x y

24

3

xy

4

3

xy