Inverse Functions

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.

If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.

If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1

A function and its inverse function are reflections of each other in the line y = x.

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.

If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1

A function and its inverse function are reflections of each other in the line y = x.

xfyabxfyba 1on point a is , then ,on point a is , If

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.

If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1

A function and its inverse function are reflections of each other in the line y = x.

xfyabxfyba 1on point a is , then ,on point a is , If

xfyxfy 1 of range theis ofdomain The

Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.

The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.

If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1

A function and its inverse function are reflections of each other in the line y = x.

xfyabxfyba 1on point a is , then ,on point a is , If

xfyxfy 1 of range theis ofdomain The

xfyxfy 1 ofdomain theis of range The

Testing For Inverse Functions

Testing For Inverse Functions

(1) Use a horizontal line test

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi y

x

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi y

x

Only has an inverse relation

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation Has an inverse function

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation Has an inverse function

OR unique.is ,asrewritten is When 2 xgyxgyyfx

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation Has an inverse function

OR unique.is ,asrewritten is When 2 xgyxgyyfx

2yx OR

Testing For Inverse Functions

(1) Use a horizontal line test

e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation Has an inverse function

OR unique.is ,asrewritten is When 2 xgyxgyyfx

2yx OR

xy NOT UNIQUE

Testing For Inverse Functions

(1) Use a horizontal line testOR

unique.is ,asrewritten is When 2 xgyxgyyfx e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation Has an inverse functionOR OR

2yx xy

NOT UNIQUE

3yx

Testing For Inverse Functions

(1) Use a horizontal line testOR

unique.is ,asrewritten is When 2 xgyxgyyfx e.g. 2xyi 3xyii y

x

y

x

Only has an inverse relation Has an inverse functionOR OR

2yx xy

NOT UNIQUE

3yx 3 xy

UNIQUE

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

yyx

xxy

2312

2312

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

yyx

xxy

2312

2312

2213132212231223

xxy

xyxyxyxyxy

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

yyx

xxy

2312

2312

2213132212231223

xxy

xyxyxyxyxy

2

23122

123

1231

xx

xx

xff

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

yyx

xxy

2312

2312

2213132212231223

xxy

xyxyxyxyxy

2

23122

123

1231

xx

xx

xff

x

xxxxx

88

46242336

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

yyx

xxy

2312

2312

2213132212231223

xxy

xyxyxyxyxy

2

23122

123

1231

xx

xx

xff

x

xxxxx

88

46242336

221323

122132

1

xx

xx

xff

If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;

xxff 1 AND xxff 1

e.g.

xxxf

2312

yyx

xxy

2312

2312

2213132212231223

xxy

xyxyxyxyxy

2

23122

123

1231

xx

xx

xff

x

xxxxx

88

46242336

221323

122132

1

xx

xx

xff

x

xxxxx

88

26662226

Restricting The Domain

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

3 e.g. xyi y

x

3xy

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

3 e.g. xyi y

x

3xy Domain: all real x

Range: all real y

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

3 e.g. xyi y

x

3xy Domain: all real x

Range: all real y

31

31 :

xy

yxf

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

3 e.g. xyi y

x

3xy Domain: all real x

Range: all real y

31

31 :

xy

yxf

Domain: all real x

Range: all real y

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

3 e.g. xyi y

x

3xy Domain: all real x

Range: all real y

31

31 :

xy

yxf

Domain: all real x

Range: all real y

Restricting The Domain

If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.

3 e.g. xyi y

x

3xy

31

xy

Domain: all real x

Range: all real y

31

31 :

xy

yxf

Domain: all real x

Range: all real y

xeyii y

x

xey

1

xeyii y

xey

Domain: all real x

Range: y > 0 1

xeyii y

x

xey

Domain: all real x

Range: y > 0

xyexf y

log:1

1

xeyii y

x

xey

Domain: all real x

Range: y > 0

xyexf y

log:1

Domain: x > 0

Range: all real y

1

xeyii y

x

xey

Domain: all real x

Range: y > 0

xyexf y

log:1

Domain: x > 0

Range: all real y

1

xeyii y

x

xey

xy logDomain: all real x

Range: y > 0

xyexf y

log:1

Domain: x > 0

Range: all real y

1

1

2xyiii y

x

2xy

2xyiii y

x

2xy Domain: all real x

Range: 0y

2xyiii y

x

2xy Domain: all real x

Range: 0y

NO INVERSE

2xyiii y

x

2xy Domain: all real x

Range: 0y

NO INVERSERestricted Domain: 0x

2xyiii y

x

2xy Domain: all real x

Range: 0y

NO INVERSERestricted Domain:

Range: 0y

0x

2xyiii y

x

2xy Domain: all real x

Range:

21

21 :

xy

yxf

0y

NO INVERSERestricted Domain:

Range: 0y

0x

2xyiii y

x

2xy Domain: all real x

Range:

21

21 :

xy

yxf

Domain:

Range:

0y

NO INVERSERestricted Domain:

Range: 0y

0x

0x0y

2xyiii y

x

2xy Domain: all real x

Range:

21

21 :

xy

yxf

Domain:

Range:

0y

NO INVERSERestricted Domain:

Range: 0y

0x

0x0y

2xyiii y

x

2xy 21

xy Domain: all real x

Range:

21

21 :

xy

yxf

Domain:

Range:

0y

NO INVERSERestricted Domain:

Range: 0y

0x

0x0y

2xyiii y

x

2xy 21

xy Domain: all real x

Range:

21

21 :

xy

yxf

Domain:

Range:

0y

NO INVERSERestricted Domain:

Range: 0y

0x

0x0y

Book 2Exercise 1A; 2, 4bdf, 7, 9, 13, 14, 16, 19