More Quarter test reviewMore Quarter test reviewSection 4.1 Section 4.1

Composite FunctionsComposite Functions

Composite FunctionsComposite Functions

Given two functions f and g, the composite function, denoted by is defined by

“f composed with g” “f composition g” “f of g”

gf ))(())(( xgfxgf

It’s kind of like if f(x) = 2x - 1, then f(2x) = 2(2x) – 1 = 4x - 1

The domain of is the set of all numbers x in the domain of g such that g(x) is in the domain of f… that is, if g(x) is not in the domain of f, then f(g(x)) is not defined. Because of this, the domain of is a subset of the domain of g; the range of is a subset of f.

(Notice that the “inside” function g in f(g(x)) is done first.)

gf

gf gf

Evaluating a CompositionEvaluating a CompositionTo evaluate , find g(x) first, then plug that answer into

f(x).

Example: Suppose that Example: Suppose that f(x) = f(x) = 22xx22 - 3 and - 3 and g(x) = g(x) = 44x. x. Find:Find:

a)

First solve for g(1). Substitute 1 in the place of x for g(x)=4x

g(1) = 4(1) = 4Use this new value in place of x in the equation f(x) = 2x2 – 3:

f(4)= 2(4)2 - 3 = 2(16) - 3 = 32 – 3 = 29

))(( xgf

)1)(( gf

b) First, find f(1): f(1) = 2(1)2 – 3 = -1 Use this value in g(x):

g(-1) = 4(-1) = -4

c) First, find f(-2): f(-2) = 2(-2)2 – 3 = 5 Use this value in f(x):

f(5) = 2(5)2 – 3 = 47

d) First, find g(-1): g(-1) = 4(-1) = -4Use this value to find g(x):

g(-4) = 4(-4) = -16

)1)(( fg

)2)(( ff

)1)(( gg

Finding a Composite Function and its Finding a Composite Function and its DomainDomain can be written as a simplified function whose

domain is based first on g, then on .

Example: Suppose that Example: Suppose that ff((xx) = ) = xx2 2 + 3+ 3x x - - 11 and and gg((xx) = 22x x + 3. Find and and their domains. + 3. Find and and their domains.

To find , substitute g(x) into f(x):

f(g) = (2x + 3)2 + 3(2x + 3) – 1 FOIL and distribute

= (4x2 + 12x + 9) + 6x + 9 - 1 Simplify

= 4x2 + 18x + 17

D: all real numbers

gf gf

gf fg

17184 2 xxgf

gf

Find the domain of the “inside” function first, then the domain of the two together…

To find , substitute f(x) into g(x):

g(f) = 2(x2 + 3x - 1) + 3 Distribute

= 2x2 + 6x – 2 + 3 Simplify = 2x2 + 6x + 1

D: all real numbers

fg

162 2 xxfg

Example:Example:Suppose that and . Find

and and their domains. To find substitute g(x) into f(x):

Simplify by multiplying numerator and denominator by x - 1

This will get rid of the fraction in the denominator

Simplify

Simplify by factoring the denominator

2

1)(

xxf

1

4)(

xxg gf ff

gf

21

41

1

4

x

xf

1

1

21

41

x

x

x

)1(24

1

x

x

)1(2

1

22

1

x

x

x

x

224

1

x

x

)1(2

1

x

xgf

To find the domain, first find the domain of g(x):

Now find the domain of :gf

Put the two together:

1

4)(

xxg 1: xD

)1(2

1

x

xgf 1: xD

1: xDomain

Suppose that and . Find and and their domains.

To find substitute f(x) into f(x):

Simplify by multiplying numerator and denominator by x + 2

This will get rid of the fraction in the denominator

Simplify

2

1)(

xxf

1

4)(

xxg gf ff

ff

2211

2

1

x

xf

2

2

2211

x

x

x

)2(21

2

x

x

421

2

x

x

52

2

x

x

52

2

x

xff

To find the domain, first find the domain of f(x):

Now find the domain of :ff

Put the two together:

2: xD

52

2

x

xff

25: xD

25,2: xDomain

2

1)(

xxf

Showing that Two Composite Showing that Two Composite Functions are EqualFunctions are Equal

…that is, prove that

Example: If and , show Example: If and , show thatthat

, for every , for every xx in the in the domain of domain of and . and .

Find by substituting g(x) into f(x):

Simplify

xxfgxgf ))(())((

43)( xxf )4(31)( xxg

xxfgxgf ))(())(( gf fg

gf

4)4(313 xgf

4)4( x

xgf x

Now find by substituting f(x) into g(x):

Simplify inside the parentheses

Multiply

fg

)443(31)( xfg

)3(31 x

xfg

fggf

“therefore”

x

More quarter test, early More quarter test, early ch 4ch 4

One-to-One and Inverse Functions

One-to-One FunctionsOne-to-One FunctionsA function is one-to-one if no y in the

range is in the image of more than one x in the domain...that is, if any two different inputs in the domain correspond to two different outputs in the range.

One-to-One Not one-to-one Not a function

Example: Determine whether Example: Determine whether the set is a one-to-one the set is a one-to-one function.function.

{(-2, 6), (-1, 3), (0, 2), (1, 5), (2, 8)}

Input Output

-2

-1

0

1

2

6

3

2

5

8

Yes, the function is one-to-one.

To determine if a graph represents To determine if a graph represents a one-to-one function:a one-to-one function:

Horizontal-line Test – If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one.

A function that is increasing on an interval I is a one-to-one function on I.

A function that is decreasing on an interval I is a one-to-one function on I.

Example: Use a horizontal line test Example: Use a horizontal line test to determine whether the graph to determine whether the graph represents a one-to-one function.represents a one-to-one function.

No, the function is not one-to-one.

Yes, the function is one-to-one.

Inverse FunctionsInverse Functions

If (x, y) is a point in f, then (y, x) is point in its inverse, f -1.

Domain of f = Range of f -1 and Range of f = Domain

of f -1.For the inverse of a function f to itself

be a function, f must be one-to-one. The graph of f and the graph of f -1

are symmetric with respect to y = x.

xffff 11To verify they are inverses…

*

Both are functions, both are one-to-one…

f f -1

Domain Domain RangeRange

Example: Find the inverse Example: Find the inverse of the following one-to-one of the following one-to-one function:function:

{(-3, -27), (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8), (3, 27)}

{(-27, -3), (-8, -2), (-1, -1), (0, 0), (1, 1), (8, 2), (27, 3)}

Example: Find the inverse of Example: Find the inverse of ff((xx) = 2) = 2xx + + 3. Identify the domain and range of 3. Identify the domain and range of ff and and ff -1-1, then graph both on the same , then graph both on the same coordinate axes.coordinate axes.1. Switch x and y:

32

32

yx

xy

2. Solve for y: 33

yx 23

32 xy

2 22

2

3

2

11 xf

y = x

f

f -1

Domain:

Range:

To verify inverse functions, To verify inverse functions, prove that prove that

Prove that the functions in the above example are inverses:

xffff 11

23

21)(

32)(1

xxf

xxf

)( 11 ffff 2321 xf 32

3212 x 3)3( x x

)(11 ffff 321 xf 23322

1 x 23

23 x x

f and f -1 are inverses

Example: Verify that the inverse Example: Verify that the inverse ofofisis 1

1)(

xxg 1

1)(1

xxg

First show that xgg 1

11)( 11

xggggg 11

1

1

xx11

x

x x

x

1

Now show that xgg 1

1

1)( 111

xggggg 1

111

x

11

1

x xx 11

11

1

1

1

1

x

x

x

g and g -1 are inverses

Example: Find the inverse Example: Find the inverse of , . Verify your result of , . Verify your result and find the domain and range of both and find the domain and range of both ff and and f f -1-1..

1

12)(

x

xxf 1x

First find1f

1

121

12

y

yx

x

xy

)1( y )1(y

12)1( yyx

12 yxxy

switch x and y

multiply both sides by y – 1

distribute x

gather y’s on same sidey2y2 xx

xyxy 12

xxy 12

factor out y

divide by x – 2

2x 2x2

1)(1

x

xxf

Example: Find the inverse Example: Find the inverse of , . Verify your result of , . Verify your result and find the domain and range of both and find the domain and range of both ff and and f f -1-1..

1

12)(

x

xxf 1x

2

1)(1

x

xxf

Can verify by graphing & check for symmetry

Now find the domain & range: If you know the domain of one, you know the range of the other…

1

12)(

x

xxf

2

1)(1

x

xxf

Domain:

Range: Range:

Domain: 1x

1y

2x

2y