Sec 5.1 and 5.2

Composites and Inverses

Math 1051 - Precalculus I

Composites and Inverses Sec 5.1 and 5.2

Sec 5.1 and 5.2 Composites and Inverses

Graph

R(x) =x2 − 1(x − 4)2

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-10

-5

5

10

Composites and Inverses Sec 5.1 and 5.2

Sec 5.1 and 5.2 Composites and Inverses

Graph

R(x) =x2 − 1(x − 4)2

-10 -5 5 10 15 20

-10

-5

5

10

Composites and Inverses Sec 5.1 and 5.2

Section 5.1 Composite Functions

(f ◦ g)(x) = f (g(x))

Composites and Inverses Sec 5.1 and 5.2

Section 5.1 Composite Functions

Given two functions f (x) and g(x) we can define the compositefunction

(f ◦ g)(x) = f (g(x))

The domain is the set of all numbers x in the domain of g(x)such that g(x) is in the domain of f (x).

Composites and Inverses Sec 5.1 and 5.2

Section 5.1 Composite Functions

Given two functions f (x) and g(x) we can define the compositefunction

(f ◦ g)(x) = f (g(x))

The domain is the set of all numbers x in the domain of g(x)such that g(x) is in the domain of f (x).

Composites and Inverses Sec 5.1 and 5.2

Section 5.1 Composite Functions

Given two functions f (x) and g(x) we can define the compositefunction

(f ◦ g)(x) = f (g(x))

The domain is the set of all numbers x in the domain of g(x)such that g(x) is in the domain of f (x).

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = 1x+3 and g(x) = − 2

x find:

(f ◦ g)(x)The domain of (f ◦ g)(x)(g ◦ f )(x)The domain of (g ◦ f )(x)

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = 1x+3 and g(x) = − 2

x find:(f ◦ g)(x)

The domain of (f ◦ g)(x)(g ◦ f )(x)The domain of (g ◦ f )(x)

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = 1x+3 and g(x) = − 2

x find:(f ◦ g)(x)The domain of (f ◦ g)(x)

(g ◦ f )(x)The domain of (g ◦ f )(x)

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = 1x+3 and g(x) = − 2

x find:(f ◦ g)(x)The domain of (f ◦ g)(x)(g ◦ f )(x)

The domain of (g ◦ f )(x)

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = 1x+3 and g(x) = − 2

x find:(f ◦ g)(x)The domain of (f ◦ g)(x)(g ◦ f )(x)The domain of (g ◦ f )(x)

Composites and Inverses Sec 5.1 and 5.2

If f (x) = 3x2 − 7 and g(x) = 2x + a, find a so that the graph of(f ◦ g)(x) crosses the y -axis at 68.

Composites and Inverses Sec 5.1 and 5.2

Decompose h(x) = (x − 5)2

Composites and Inverses Sec 5.1 and 5.2

Section 5.2 Inverse Functions

Remember a function is something that takes in one inputand gives exactly one outputThe graph of a function passes the vertical line test

One-to-one Function: A function is one-to-one if two differentinputs always give two different outputs:

If x1 6= x2 then f (x1) 6= f (x2)

Composites and Inverses Sec 5.1 and 5.2

Section 5.2 Inverse Functions

Remember a function is something that takes in one inputand gives exactly one output

The graph of a function passes the vertical line test

One-to-one Function: A function is one-to-one if two differentinputs always give two different outputs:

If x1 6= x2 then f (x1) 6= f (x2)

Composites and Inverses Sec 5.1 and 5.2

Section 5.2 Inverse Functions

Remember a function is something that takes in one inputand gives exactly one outputThe graph of a function passes the vertical line test

One-to-one Function: A function is one-to-one if two differentinputs always give two different outputs:

If x1 6= x2 then f (x1) 6= f (x2)

Composites and Inverses Sec 5.1 and 5.2

Section 5.2 Inverse Functions

Remember a function is something that takes in one inputand gives exactly one outputThe graph of a function passes the vertical line test

One-to-one Function: A function is one-to-one if two differentinputs always give two different outputs:

If x1 6= x2 then f (x1) 6= f (x2)

Composites and Inverses Sec 5.1 and 5.2

Section 5.2 Inverse Functions

Remember a function is something that takes in one inputand gives exactly one outputThe graph of a function passes the vertical line test

One-to-one Function: A function is one-to-one if two differentinputs always give two different outputs:

If x1 6= x2 then f (x1) 6= f (x2)

Composites and Inverses Sec 5.1 and 5.2

Section 5.2 Inverse Functions

Remember a function is something that takes in one inputand gives exactly one outputThe graph of a function passes the vertical line test

One-to-one Function: A function is one-to-one if two differentinputs always give two different outputs:

If x1 6= x2 then f (x1) 6= f (x2)

Composites and Inverses Sec 5.1 and 5.2

Horizontal Line Test

We can test whether a function is one-to-one using the“horizontal line test”

Composites and Inverses Sec 5.1 and 5.2

Horizontal Line Test

We can test whether a function is one-to-one using the“horizontal line test”

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f (x) = x2

Composites and Inverses Sec 5.1 and 5.2

Horizontal Line Test

We can test whether a function is one-to-one using the“horizontal line test”

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f (x) = x2, x ≥ 0

Composites and Inverses Sec 5.1 and 5.2

Inverse of a Function

The inverse of a function f is a correspondence from the rangeof f back to the domain.

The inverse of f undoes whatever f did, and is denoted f−1(x)...

For example, the inverse of f (x) = x2 is f−1(x) =√

x

Be careful, even though x−1 = 1x , f−1(x) 6= 1

f (x)

Composites and Inverses Sec 5.1 and 5.2

Inverse of a Function

The inverse of a function f is a correspondence from the rangeof f back to the domain.

The inverse of f undoes whatever f did, and is denoted f−1(x)...

For example, the inverse of f (x) = x2 is f−1(x) =√

x

Be careful, even though x−1 = 1x , f−1(x) 6= 1

f (x)

Composites and Inverses Sec 5.1 and 5.2

Inverse of a Function

The inverse of a function f is a correspondence from the rangeof f back to the domain.

The inverse of f undoes whatever f did, and is denoted f−1(x)...

For example, the inverse of f (x) = x2 is f−1(x) =√

x

Be careful, even though x−1 = 1x , f−1(x) 6= 1

f (x)

Composites and Inverses Sec 5.1 and 5.2

Inverse of a Function

The inverse of a function f is a correspondence from the rangeof f back to the domain.

The inverse of f undoes whatever f did, and is denoted f−1(x)...

For example, the inverse of f (x) = x2 is f−1(x) =√

x

Be careful, even though x−1 = 1x , f−1(x) 6= 1

f (x)

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = {(2,3), (5,7), (8,−1)} what is the inverse?

Ans: f−1(x) = {(3,2), (7,5), (−1,8)}

Composites and Inverses Sec 5.1 and 5.2

Given f (x) = {(2,3), (5,7), (8,−1)} what is the inverse?

Ans: f−1(x) = {(3,2), (7,5), (−1,8)}

Composites and Inverses Sec 5.1 and 5.2

To show that two functions f and g are inverses, we need toshow

f (g(x)) = x and g(f (x)) = x

For example,

f (x) =3

2− x

g(x) =2x − 3

x

Composites and Inverses Sec 5.1 and 5.2

To show that two functions f and g are inverses, we need toshow

f (g(x)) = x and g(f (x)) = x

For example,

f (x) =3

2− x

g(x) =2x − 3

x

Composites and Inverses Sec 5.1 and 5.2

Graphing inverses

If f (x) and g(x) are inverses of each other and the point (a,b)is on the graph of f (x), then the point (b,a) is on the graph ofg(x).

For example, if f (x) = x3 + 3 and g(x) = f−1(x) = 3√

x − 3

f (1) = (1)3 + 3 = 4

g(4) = 3√

4− 3 = 1

Composites and Inverses Sec 5.1 and 5.2

Graphing inverses

If f (x) and g(x) are inverses of each other and the point (a,b)is on the graph of f (x), then the point (b,a) is on the graph ofg(x).

For example, if f (x) = x3 + 3 and g(x) = f−1(x) = 3√

x − 3

f (1) = (1)3 + 3 = 4

g(4) = 3√

4− 3 = 1

Composites and Inverses Sec 5.1 and 5.2

Graphing inverses

If f (x) and g(x) are inverses of each other and the point (a,b)is on the graph of f (x), then the point (b,a) is on the graph ofg(x).

For example, if f (x) = x3 + 3 and g(x) = f−1(x) = 3√

x − 3

f (1) = (1)3 + 3 = 4

g(4) = 3√

4− 3 = 1

Composites and Inverses Sec 5.1 and 5.2

Graphing inverses

If f (x) and g(x) are inverses of each other and the point (a,b)is on the graph of f (x), then the point (b,a) is on the graph ofg(x).

For example, if f (x) = x3 + 3 and g(x) = f−1(x) = 3√

x − 3

f (1) = (1)3 + 3 = 4

g(4) = 3√

4− 3 = 1

Composites and Inverses Sec 5.1 and 5.2

If we graph the functions we will see they are symmetric aboutthe line y = x :

Composites and Inverses Sec 5.1 and 5.2

If we graph the functions we will see they are symmetric aboutthe line y = x :

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f (x) = x3 + 3

Composites and Inverses Sec 5.1 and 5.2

If we graph the functions we will see they are symmetric aboutthe line y = x :

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f−1(x) = 3√

x − 3

Composites and Inverses Sec 5.1 and 5.2

If we graph the functions we will see they are symmetric aboutthe line y = x :

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

y = x

Composites and Inverses Sec 5.1 and 5.2

The inverse of f (x) =√

x

Composites and Inverses Sec 5.1 and 5.2

The inverse of f (x) =√

x

-4 -2 2 4

-4

-2

2

4

f (x) =√

x

Composites and Inverses Sec 5.1 and 5.2

The inverse of f (x) =√

x

-4 -2 2 4

-4

-2

2

4

f (x) =√

x

Composites and Inverses Sec 5.1 and 5.2

The inverse of f (x) =√

x

-4 -2 2 4

-4

-2

2

4

f−1(x) = x2, x ≥ 0

Composites and Inverses Sec 5.1 and 5.2

In general, to solve for the inverse algebraically, exchange xand y and then solve for y

For example, if f (x) = x3 + 1 find the inverse

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Composites and Inverses Sec 5.1 and 5.2

In general, to solve for the inverse algebraically, exchange xand y and then solve for y

For example, if f (x) = x3 + 1 find the inverse

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Composites and Inverses Sec 5.1 and 5.2

In general, to solve for the inverse algebraically, exchange xand y and then solve for y

For example, if f (x) = x3 + 1 find the inverse

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 4x+2

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 4x+2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f (x) =4

x + 2

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 4x+2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f (x) =4

x + 2

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 4x+2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

f−1(x) =4x− 2

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 2x−3x+4

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 2x−3x+4

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-20

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10

20

f (x) =2x − 3x + 4

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 2x−3x+4

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10

20

f (x) =2x − 3x + 4

Composites and Inverses Sec 5.1 and 5.2

Find the inverse of f (x) = 2x−3x+4

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-20

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10

20

f−1(x) =−4x − 3

x − 2

Composites and Inverses Sec 5.1 and 5.2

Read section 5.3 for Wednesday.

Composites and Inverses Sec 5.1 and 5.2