Pc Inverse Functions

Inverse Functions By MATH PROJECT

To have an inverse, a function must be oneto-one. This means for each x there is one and only one y and for each y there is one and only one x.

The Horizontal Line Test is used to test a function for an inverse. If a horizontal line is drawn and intersects the graph at more than one point, the function does not have an inverse. The function has an inverse if the horizontal line intersects the graph at only one point.

The function y=x2 fails the Horizontal Line Test and therefore doesn’t have an inverse.

The function y=x+2 passes the Horizontal Line Test and has an inverse.

To find the inverse… Use the Horizontal Line Test. Replace f(x) with y. Exchange x and y then solve for y. Replace y with f-1(x). Check: f ° f-1, f-1 ° f, and the graph.

f ( x)  3x Replace f(x) with y. Exchange x and y. Solve for y. Replace y with f-1(x). Check.

y  3x x  3y x y 3

x f (x)  3 1

Check by graphing.

x f (x )  f ( x)  3x 3 The graph of the inverse is the graph of the function reflected across the line y=x. 1

Check with composition of functions. f f-1

f-1 f

x f ( f ( x ))  f ( ) 3 x x f ( )  3( ) 3 3 1

1

f ( f ( x))  x 1

1

1

f ( f ( x))  f (3 x) (3x ) f (3 x)  3 1

1

f ( f ( x))  x 1

f ( f ( x))  f ( f ( x))  x

f ( x)  x  1 3

y  x 1 3 x  y 1 3

Replace f(x) with y. Exchange x and y. Solve for y. Replace y with f-1(x). Check.

x 1  y 3

1

3

x 1  y

f ( x) 

3

x 1

Check by graphing.

f ( x)  x  1 3

1

f ( x) 

3

x 1

The graph of f(x) and the graph of f-1(x) should reflect over the line y=x. y=f(x)

y=x y=f-1(x)

Check with composition of functions. f f-1 1

f-1 f

f ( f ( x))  f ( x  1) 3

f ( x  1)  ( x  1)  1 3

3

3

1

f ( f ( x))  x 1

1

1

f ( f ( x))  f ( x  1) 1

3

f ( x  1)  ( x  1)  1 3

3

1

f ( f ( x))  x 1

f ( f ( x))  f ( f ( x))  x

3

f ( x) | x- 2|

either  x  2orx  2 The absolute value function does not have inverse because it is not one-to-one and fails the Horizontal Line Test. For the absolute value function to have an inverse, there needs to be a restricted domain.

f ( x) | x- 2|

either  x  2orx  2 Replace f(x) with y.

y  x2

Exchange x and y.

x  y2

Solve for y. Replace y with f-1(x).

x2 y 1 f ( x)  x  2, x  2

Check by graphing & composition of functions.