11 x1 t02 08 inverse functions (2012)

Inverse Relations

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x 3inverse relation is x y y

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

3inverse relation is x y y

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

3inverse relation is x y y

The range of the relation is the domain of its inverse relation

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

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

3inverse relation is x y y

The range of the relation is the domain of its inverse relation

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

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

2e.g. y x

3inverse relation is x y y

The range of the relation is the domain of its inverse relation

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

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

2e.g. y x

3inverse relation is x y y

The range of the relation is the domain of its inverse relation

domain: all real xrange: 0y

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

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

2e.g. y x

3inverse relation is x y y

The range of the relation is the domain of its inverse relation

domain: all real xrange: 0y

2inverse relation: x y

Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)

3e.g. y x x

The domain of the relation is the range of its inverse relation

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

2e.g. y x

3inverse relation is x y y

The range of the relation is the domain of its inverse relation

domain: all real xrange: 0y

2inverse relation: x ydomain: 0x

range: all real y

Inverse Functions

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy NOT UNIQUE

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy NOT UNIQUE

y

x

3xyii

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy NOT UNIQUE

y

x

3xyii

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy NOT UNIQUE

y

x

3xyii

Has an inverse function

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy NOT UNIQUE

y

x

3xyii

Has an inverse function

OR3yx

3 xy

Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.

Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y

x

Only has an inverse relation

OR2yx

xy NOT UNIQUE

y

x

3xyii

Has an inverse function

OR3yx

3 xy UNIQUE

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. 2

2xf xx

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. 2

2xf xx

2 22 2

x yy xx y

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. 2

2xf xx

2 22 2

x yy xx y

2 22 2

1 2 22 21

y x yxy x yx y x

xyx

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. 2

2xf xx

2 22 2

x yy xx y

2 22 2

1 2 22 21

y x yxy x yx y x

xyx

1

22 22

212

xxf f x

xx

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. 2

2xf xx

2 22 2

x yy xx y

2 22 2

1 2 22 21

y x yxy x yx y x

xyx

1

22 22

212

xxf f x

xx

2 4 2 42 2

44

x xx xx

x

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. 2

2xf xx

2 22 2

x yy xx y

2 22 2

1 2 22 21

y x yxy x yx y x

xyx

1

22 22

212

xxf f x

xx

2 4 2 42 2

44

x xx xx

x

1

2 2 21

2 2 21

xxf f x

xx

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. 2

2xf xx

2 22 2

x yy xx y

2 22 2

1 2 22 21

y x yxy x yx y x

xyx

1

22 22

212

xxf f x

xx

2 4 2 42 2

44

x xx xx

x

1

2 2 21

2 2 21

xxf f x

xx

2 2 2 22 2 2 244

x xx xx

x

(ii) Draw the inverse relation

y

x

(ii) Draw the inverse relation

y

x

(ii) Draw the inverse relation

y

x

(ii) Draw the inverse relation

y

x

Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj