7.1 – Operations on Functions

94
7.1 – Operations on Functions

description

7.1 – Operations on Functions. OperationDefinition. OperationDefinition Sum. OperationDefinition Sum( f + g )( x ). OperationDefinition Sum( f + g )( x ) = f ( x ) + g ( x ). OperationDefinition Sum( f + g )( x ) = f ( x ) + g ( x ) - PowerPoint PPT Presentation

Transcript of 7.1 – Operations on Functions

Page 1: 7.1 – Operations on Functions

7.1 – Operations on Functions

Page 2: 7.1 – Operations on Functions

Operation Definition

Page 3: 7.1 – Operations on Functions

Operation Definition

Sum

Page 4: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x)

Page 5: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Page 6: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference

Page 7: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) =

Page 8: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Page 9: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product

Page 10: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) =

Page 11: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Page 12: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Quotient f (x) =

g

Page 13: 7.1 – Operations on Functions

Operation Definition

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Quotient f (x) = f(x)

g g(x)

Page 14: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

Page 15: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x)

Page 16: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

Page 17: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

Page 18: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3)

Page 19: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3)

Page 20: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

Page 21: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x + 6

Page 22: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x)

Page 23: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

Page 24: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

Page 25: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3)

Page 26: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3)

Page 27: 7.1 – Operations on Functions

Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for

f(x) gand g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)= (2x – 3) + (4x + 9)= 6x – 6

(f – g)(x) = f(x) – g(x)= (2x – 3) – (4x + 9)

Page 28: 7.1 – Operations on Functions

Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for

f(x) gand g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)= (2x – 3) + (4x + 9)= 6x – 6

(f – g)(x) = f(x) – g(x)= (2x – 3) – (4x + 9)

Page 29: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3) – (4x + 9)

= 2x – 3 – 4x

Page 30: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3) – (4x + 9)

= 2x – 3 – 4x – 9

Page 31: 7.1 – Operations on Functions

Ex. 1

Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x) for f(x) g

and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

(f + g)(x) = f(x) + g(x)

= (2x – 3) + (4x + 9)

= 6x – 6

(f – g)(x) = f(x) – g(x)

= (2x – 3) – (4x + 9)

= 2x – 3 – 4x – 9

= -2x – 12

Page 32: 7.1 – Operations on Functions

(f · g)(x)

Page 33: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

Page 34: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

Page 35: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)

Page 36: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)

Page 37: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

Page 38: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

Page 39: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

Page 40: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x)

g

Page 41: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x) = f(x)

g g(x)

Page 42: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x) = f(x)

g g(x)

= 2x – 3

4x + 9

Page 43: 7.1 – Operations on Functions

(f · g)(x) = f(x) · g(x)

= (2x – 3)(4x + 9)

= 8x2 + 18x – 12x – 27

= 8x2 + 6x – 27

f (x) = f(x)

g g(x)

= 2x – 3

4x + 9

*Factor & Simplify if possible!

Page 44: 7.1 – Operations on Functions

Composite Function

Page 45: 7.1 – Operations on Functions

Composite Function

- taking the function

Page 46: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

Page 47: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)]

Page 48: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Page 49: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

Page 50: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

Page 51: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

Page 52: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

= f[x2 + x – 1]

Page 53: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

= f[x2 + x – 1]

Page 54: 7.1 – Operations on Functions

Composite Function

- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]

= f[x2 + x – 1]

Page 55: 7.1 – Operations on Functions

Composite Function- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]= f(x2 + x – 1)= (x2 + x – 1) + 3

Page 56: 7.1 – Operations on Functions

Composite Function- taking the function of a function

[f °g(x)] = f[g(x)]

Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x2 + x – 1.

[f °g(x)] = f[g(x)]= f(x2 + x – 1)= (x2 + x – 1) + 3= x2 + x + 2

Page 57: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)]

Page 58: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

Page 59: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

Page 60: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

Page 61: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

Page 62: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

Page 63: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

Page 64: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2

Page 65: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2

Page 66: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3)

Page 67: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3)

Page 68: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

Page 69: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

Page 70: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

= x2 + 6x + 9 + x + 3 – 1

Page 71: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

= x2 + 6x + 9 + x + 3 – 1

= x2 + 7x + 11

Page 72: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

Page 73: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] =

Page 74: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] =

Page 75: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] =

Page 76: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1

[g°f(x)] = g[f(x)]

= g(x + 3)

= (x + 3)2 + (x + 3) – 1

= (x + 3)(x + 3) + (x + 3) – 1

= x2 + 6x + 9 + x + 3 – 1

= x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

g[f(5)] = g[4(5)]

Page 77: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20)

Page 78: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20)

Page 79: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20) = 2(20) – 1

Page 80: 7.1 – Operations on Functions

f(x) = x + 3 and g(x) = x2 + x – 1 [g°f(x)] = g[f(x)]

= g(x + 3)= (x + 3)2 + (x + 3) – 1= (x + 3)(x + 3) + (x + 3) – 1= x2 + 6x + 9 + x + 3 – 1 = x2 + 7x + 11

Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

= g(20) = 2(20) – 1 = 39

Page 81: 7.1 – Operations on Functions

7.3 – Square Root Functions & Inequalities

Page 82: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

Page 83: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

Page 84: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Page 85: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

Page 86: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

Page 87: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

Page 88: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

y = √ -4+ 4

Page 89: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

y = √ -4+ 4

y = 0

Page 90: 7.1 – Operations on Functions

Ex. 1 Identify the domain & range of each function.

a. y = √ x + 4

x + 4 = 0

x = -4

Domain: { x | x > -4}

y = √ x + 4

y = √ -4+ 4

y = 0

Range: { y | y > 0}

Page 91: 7.1 – Operations on Functions

Ex. 2 Graph each function. State the domain & range.

a. y = √ x + 4

Domain: { x | x > -4}, Range: { y | y > 0}

Graph: Y=

2nd, x2

x + 4)

Zoom:6

2nd Graph

Plot at least 3 points of curve

(x & y ints. & one other pt.)

Page 92: 7.1 – Operations on Functions

x y

-4 0

-3 1

0 2

Page 93: 7.1 – Operations on Functions

Ex. 3 Graph each inequality

a. y <√ x + 4

Graph: Y= Cursor left to \

Press “Enter” until

(If > make it )

2nd, x2

x + 4)

Zoom:6

2nd Graph

Plot at least 3 points of curve

(x & y ints. & one other pt.)

Page 94: 7.1 – Operations on Functions

x y

-4 0

-3 1

0 2