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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

• 7.1 Operations on Functions

• OperationDefinition

• OperationDefinitionSum

• OperationDefinitionSum(f + g)(x)

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)Difference

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)Difference(f g)(x) =

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)Difference(f g)(x) = f(x) g(x)

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)Difference(f g)(x) = f(x) g(x)Product

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)Difference(f g)(x) = f(x) g(x)Product(f g)(x) =

• OperationDefinitionSum(f + g)(x) = f(x) + g(x)Difference(f g)(x) = f(x) g(x)Product(f g)(x) = f(x) g(x)

• OperationDefinitionSum(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

• OperationDefinitionSum(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) gand 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) gand 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) gand 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) gand 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) gand 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) gand 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) gand 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) 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

• 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)

• 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)

• 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)

• 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)

• 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)

• 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)

• 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)

• 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)= 2x 3 4x

• 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)= 2x 3 4x 9

• 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)= 2x 3 4x 9= -2x 12

• (f g)(x)

• (f g)(x) = f(x) g(x)

• (f g)(x) = f(x) 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)= 8x2 + 18x 12x 27

• (f g)(x) = f(x) g(x)= (2x 3)(4x + 9)= 8x2 + 18x 12x 27 = 8x2 + 6x 27

• (f g)(x) = f(x) g(x)= (2x 3)(4x + 9)= 8x2 + 18x 12x 27 = 8x2 + 6x 27 f (x) g

• (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)

• (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

• (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!

• Composite Function

• 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 [gf(x)] for the functions f(x) = x + 3 and g(x) = x2 + x 1.

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

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

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

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

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

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

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

Ex. 2 Find [f g(x)] and [gf(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

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

Ex. 2 Find [f g(x)] and [gf(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

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)]

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)= (x + 3)2

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)= (x + 3)2

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)= (x + 3)2 + (x + 3)

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)= (x + 3)2 + (x + 3)

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x + 3)= (x + 3)2 + (x + 3) 1

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(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) = x2 + x 1 [gf(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

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(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

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(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)].

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(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)] =

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(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)] =

• f(x) = x + 3 and g(x) = x2 + x 1 [gf(x)] = g[f(x)]= g(x