7.1 – Operations on Functions. OperationDefinition.

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Presentation transcript:

7.1 – Operations on Functions

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) Difference

OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) =

OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x)

OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product

OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product(f · g)(x) =

OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product(f · g)(x) = f(x) · g(x)

OperationDefinition 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

OperationDefinition 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)

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

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)

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)

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)

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)

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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

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

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

(f · g)(x)

(f · g)(x) = f(x) · g(x)

= (2x – 3)

(f · g)(x) = f(x) · g(x) = (2x – 3)

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

(f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x x – 12x – 27

(f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x x – 12x – 27 = 8x 2 + 6x – 27

(f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x x – 12x – 27 = 8x 2 + 6x – 27 f (x) g

(f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x x – 12x – 27 = 8x 2 + 6x – 27 f (x) = f(x) g g(x)

(f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x x – 12x – 27 = 8x 2 + 6x – 27 f (x) = f(x) g g(x) = 2x – 3 4x + 9

(f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x x – 12x – 27 = 8x 2 + 6x – 27 f (x) = f(x) g g(x) = 2x – 3 4x + 9 *Factor & Simplify if possible!

Composite Function

- taking the function

Composite Function - taking the function of a function

Composite Function - taking the function of a function [f °g(x)]

Composite Function - taking the function of a function [f °g(x)] = f[g(x)]

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) = x 2 + x – 1.

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) = x 2 + x – 1. [f °g(x)] = f[g(x)]

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) = x 2 + x – 1. [f °g(x)] = f[g(x)]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 7x + 11

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] =

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] =

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] =

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 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)

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 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)

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 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

f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x x + 3 – 1 = x 2 + 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

7.3 – Square Root Functions & Inequalities

Ex. 1 Identify the domain & range of each function. a. y = √ x + 4

Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0

Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4

Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4}

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

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

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

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 = √ y = 0

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 = √ y = 0 Range: { y | y > 0}

Ex. 2 Graph each function. State the domain & range. a. y = √ x + 4 Domain: { x | x > -4}, Range: { y | y > 0} Graph: Y= 2 nd, x 2 x + 4) Zoom:6 2 nd Graph Plot at least 3 points of curve (x & y ints. & one other pt.)

xy

Ex. 3 Graph each inequality a. y <√ x + 4 Graph: Y= Cursor left to \ Press “Enter” until (If > make it ) 2 nd, x 2 x + 4) Zoom:6 2 nd Graph Plot at least 3 points of curve (x & y ints. & one other pt.)

xy