Operation of Functions and Inverse Functions Sections 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. 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) = x 2 -3x + 1 and g(x) = 4x + 5. Find (f + g)(x).

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) = x 2 -3x +1 and g(x) = 4x +5, then (f-g)(x) = (x 2 -3x + 1) – (4x +5) X 2 -3x +1 -4x -5 Combine like terms X 2 -7x -4 Simplify

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 3x 2 Outer -2x Inner -15x Last 10 3x 2 -2x -15x +10 = 3x 2 -17x + 10

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) = x 2 + 6x + 9 and g(x) = x +3.Use synthetic division __-3 -9___ Answer: x + 3

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)}

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 Y = x -4 Step4: replace y with g -1 (x); g -1 (x) = x -4

You try it!! F(x) = x 2 -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) 5.Find the inverse of {(3,7), (9,0), (8,6), (5,8)} 6.Find the inverse function for f(x) = 3x -2