7.1 – Operations on Functions

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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