Duplex Fractions, f(x), and composite functions

[f(x) = Find f-1(x)]

A. [3x – 5 ]B. [3x – 15 ]C. [1.5x – 7.5 ]D. [Option 4]

53

2x

[h(x)= x3 - 5. Find h-1(x) ]

A. [Option 1]B. [x3 + 5 ]C. [Option 3]D. [Option 4]

3 5x

3

5x

53 x

453

2x

4

5

3

2x

5

4*

3

2x

15

8x

453

2x

5*3

4*2x

15

8x

y

x

2153

5

)15(3

25 yx

9

2

45

10 xyxy

y342

y3412

2

3

4

6 yy

[Simplify]

25.00.1

10125

[Simplify]

10.1

2918

Which are wrong?

A) -32 = -9B) (-3)2 = 9C) x2 when x = -3 is -9

D) 2(4)2 = 64

E) 5 – (-2)2 = 9

(-3)2 = 9

2(16) = 32

5 - 4 = 1

f(x)

• It means to simplify when x = ( ) Ex: f(x) = 3x + 1. Find f(-2) means 3(-2) + 1 = -5 Find f(a) means 3(a) + 1 = 3a + 1 Find f(a + 3) means 3(a+3) + 1 = 3a + 10

•Or find y when x is ( )•f(1) is when x is 1 so 2•f(-4) is -2

[Find f(3)]

A. [3]B. [-1]C. [0]D. [2]

[Find the value of f(4) and g(-10) if f(x)=-8x-8 and g(x)=2x2-22x]

A. [-24, -2208]B. [-40, 420]C. [80, 8]D. [-16, 102]

©1999 by Design Science, Inc. 13

Composition of functions• Composition of functions is the successive

application of the functions in a specific order.

• Given two functions f and g, the composite function is defined by and is read “f of g of x.”

• The domain of is the set of elements x in the domain of g such that g(x) is in the domain of f.

– Another way to say that is to say that “the range of function g must be in the domain of function f.”

f g f g x f g x

f g

©1999 by Design Science, Inc. 14

f g

A composite function

x

g(x)

f(g(x))

domain of grange of f

range of g

domain of f

g

f

©1999 by Design Science, Inc. 15

g x

A different way to look at it…

FunctionMachine

x f g x

FunctionMachine

gf

f(x) = 3x + 2 g(x) = 2x - 5

f o g(3) f(g(3)) = f(2(3) – 5) = f(1) = 3(1) + 2 = 5Plug 3 into g, get the answer, give it to f

g o f(3) g(f(3)) = g(3(3) + 2) = g(11) = 2(11) – 5 = 17Plug 3 into f, get the answer, give it to g

f(x) = 3x2 - 1 g(x) = x - 5

Find f o g (-2) and g o f(-2)

f o g (-2) = f(g(-2)) = f(-2-5) = f(-7) = 3(-7)2 – 1 = 146

go f (-2) = g(f(-2)) = g(3(-2)2 - 1) = g(11)= 6

f(x) = 3x + 2 g(x) = 2x - 5

f o g(x) f(g(x)) = f(2x – 5) = 3(2x-5)+ 2 = 6x-15+2 = 6x-13Plug x into g, get equation, give it to f

g o f(x) g(f(x)) = g(3x+2) = 2(3x+2)-5 = 6x +4-5= 6x-1Plug x into f, get equation, give it to g

Two functions, f(x) and g(x), are inverses if and only if fog(x)=x and g o f(x)=x

Ex: f(x) = 3x + 2 g(x)= 3

2x

f o g(x) = f( ) = 3 + 23

2x

3

)2( x

= x – 2 + 2 = xg o f(x) = = 3x/3 = x

3

223 x

Are the following functions inverses?

Answer: Yes!4

3)(

43)(

xxg

xxf

Function

A function is a mathematical “rule” that for each “input” (x-value) there is one and only one “output” (y – value). <vertical line test>

A function has a domain (input or x) and a range (output or y)

A one-to-one function has only one x for each y! <vertical and horizontal line test>

Examples of a Function

{ (2,3) (4,6) (7,8)(-1,2)(0,4)} 4

-2

1

8

-4

2

4

-2

1

8

-4

2

Non – Examples of a Function

{(1,2) (1,3) (1,4) (2,3)}

Vertical Line Test – if it passes through the graph more than once then it is NOT a function.

You Do: Is it a Function?

1.{(2,3) (2,4) (3,5) (4,1)}

2.{(1,2) (-1,3) (5,3) (-2,4)}

3. 4.

5.

0

-3

4

1

-5

9