Inverse Functions

Not all functions have inverse functions. A function has an inverse function

if and only ifthe function is a one to one relation.

Determine whether each of the following functions has an inverse function. Given reasons for your answers.

Yes, the function has an inverse function.

Determine whether each of the following functions has an inverse function. Given reasons for your answers.

N0, the function does not has an inverse function (not an one –to-one relation)

2

3

4

5

Determine whether each of the following functions has an inverse function. Given reasons for your answers.

Yes, the function has an inverse function.

0

2

4

6

A

0

1

2

3

B

Determine whether each of the following functions has an inverse function. Given reasons for your answers.

0

1

2

3

4

A

0

2

4

6

B

N0, the function does not has an inverse function (not an one –to-one relation)

)3(1f )(1 xf

The function f is defined as f(x) = 2x – 5. Find (a) (b)

3 2x - 5

2x 3+5

x

8

2

84

4)3(1 f

x 2x - 5

2x x+5

x 2

5x

2

5)(1

x

xf

)3(1f )(1 xf

The function f is defined as f(x) = 2x – 5. Find (a) (b)

yf )3(1let 3)( yf

352 y532 y

82 y28y

4y4)3(1 f

yxf )(1let xyf )(

xy 5252 xy

2

5

xy

2

5)(1

x

xf

xxf 49)( (a)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

y 9 – 4x- 4x y - 9

x 4

9

y

4

9)(1

xxf

4

9

y

4

9

y

4

9 y

xxf

10)( (b)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

xxf

10)(1

yx

10

10

xy

1

xy

10

0, x

3

2)(

x

xxf(c)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

x

xxf

1

23)(1

3

2

x

x

2x)3( xy yxy 3

1, x

x23 yxy

23 y xyx )1( yx

y

y

y

1

23 x23 y

43)( xxf(d)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

3

4)(1

x

xf

43 x

x34y

y

3

4y x

14

1)( xxf(e)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

)1(4)(1 xxf

14

1x

x4

11y

y

)1(4 y x

2

3)(

xxf(f)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

23

)(1

xxf

2

3

x

3

2xy

1

y

y

3 2x

23

y

x

0, x

4

32)(

x

xxf(g)

,

Find the inverse function fˉ¹(x) for each of the function f(x) below.

x

xxf

2

34)(1

4

32

x

x

32 xyxy 4

y

34 y xyx 2

34 y )2( yx

2, x

3xy

y

y

2

34 x

)(1 xf (a)

2. Given that f(x) = x – 5 and g(x) =

5)(1 xxf

1

2

x

x

x5y

y

Find

and )3(1f

5x

(a)

53)3(1 f

8

)(1 xg (b)

2. Given that f(x) = x – 5 and g(x) =

x

xxg

1

2)(1

1

2

x

x

2xyxy

y

2 yxy x2 y xyx

1, x

y

y

1

2 x

Find

and )2(1g

1

2

x

x

2 y )1( yx

(b) x

xxg

1

2)(1

21

22)2(1

f

1

4

4