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Operation of Functions and Inverse Functions

Operation of Functions and Inverse Functions. Sections 7.7- 7.8 Finding the sum, difference, product, and quotient of functions and inverse of functions. Operation of Functions. Let f(x) and g(x) by any two functions.

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Operation of Functions and Inverse Functions

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  1. Operation of Functions and Inverse Functions Sections 7.7- 7.8 Finding the sum, difference, product, and quotient of functions and inverse of functions

  2. Operation of Functions • Let f(x) and g(x) by any two functions. • You can add, subtract, multiply, and divide functions according to the following rules: • Sum- (f + g)(x) = f(x) + g(x). • Ex. If f(x) = x +2 and g(x) = 3x, then (f + g) (x) = (x + 2) + 3x) = 4x +2 • You try one: if f(x) = x2 -3x + 1 and g(x) = 4x + 5. Find (f + g)(x).

  3. Difference • (f –g)(x) = f(x) –g(x) • If f(x) = x + 2 and g(x) = 3x. Then (f-g)(x) = (x-2) -3x = x -3x -2 = -2x +2. • If f(x) = x2 -3x +1 and g(x) = 4x +5, then (f-g)(x) = (x2-3x + 1) – (4x +5) • X2 -3x +1 -4x -5 Combine like terms • X2 -7x -4 Simplify

  4. Product • (f * g)(x) = f(x) * g(x) • If f(x ) = x-5 and g(x) = 3x -2, find f(x)*g(x). • (x-5)(3x-2) Use FOIL • First 3x2 • Outer -2x • Inner -15x • Last 10 • 3x2 -2x -15x +10 = 3x2 -17x + 10

  5. Quotient • (f/g)(x) = f(x)/g(x) • Find f(x)/ g(x) if f(x) = x + 9 and g(x) = x -9 • (x+9) / (x -9) Can not simplify further EX2: Find f(x)/g(x) when f(x) = x2 + 6x + 9 and g(x) = x +3.Use synthetic division • -3 1 6 9 • __-3 -9___ • 1 3 0 • Answer: x + 3

  6. Inverse of functions • Recall that a relation is a set of ordered pairs. The inverse relation is the set of ordered pairs obtained by reversing the coordinates of each original ordered pair. The domain become the range and the range became the domain. • Ex. Find the inverse of the following relation: • {(1,8), (8,3),(3,4), (4,1), (5,6), (6,7)} • Answers: {(8,1), (3,8), (4,3), (1,4), (6,5) , (7,6)}

  7. Finding an inverse function • Write the inverse of a function f(x) as f-1 (x). • Find the inverse of g(x) = x + 4 • Step1: replace g(x) with y; y = x + 4 • Step2: interchange x and y; x = y + 4 • Step3: Solve for y; x -4 = y + 4 -4 • Y = x -4 • Step4: replace y with g-1 (x); g-1 (x) = x -4

  8. You try it!! F(x) = x2 -2x + 1 g(x) = x-1 1.Find (f +g) (x) 2.Find (f –g) (x) 3.Find (f *g) (x) 4. Find (f/g)(x) • Find the inverse of {(3,7), (9,0), (8,6), (5,8)} • Find the inverse function for f(x) = 3x -2

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