2.6 Operations and Composition

Quiz Let f(x) = 3x – 1, g(x) = x2-4, then (f + g)(x) = ______. (write the function expression )

Operations on Functions Given two functions f and g, suppose Domain of f is Df, Domain of g is Dg F(x) Sum (f + g)(x) = f(x) + g(x) Domain: Df intersect Dg Difference (f - g)(x) = f(x) – g(x) Product (fg)(x) = f(x)g(x) Quotient (f/g)(x) = f(x) / g(x) Df ∪ Dg, where x such that g(x) ≠ 0

Operations on Functions f(x) = 3x -1 g(x) = x2 -4 h(x) = x + 2 p(x) = (x + 3)0.5 Evaluate: (f+g)(6); Find (fh)(x); Find (g - f)(x). Is this the same as (f - g)(x)? Find (g / h)(x) Find (h - p)(x) Evaluate: (h - p)(1); (h - p)(-4)

Operations on Functions Discussion 1, h(x) = x2 + 2x (a) Find h(x + 1) (a) Find h(x) + h(1) 2, f(x) = 3x + 5 (a) Find f(1/2) (b) Find f(1) / f(2)

Operations on Functions Example If possible, use the given representations of functions f and g to evaluate (f + g)(4), (f - g)(-2), (fg)(1), and (f/g)(0) f(x) = 2x + 1 g(x) = √ x Step 1: Find domains of f and g Step 2: Make sure the input is in the domain of f and g. If it is a quotient operation, make sure the denominator doesn’t equal to 0. Step 3: Find corresponding expressions of operations Step 4: Evaluate new generated functions

Difference Quotients Consider a nonlinear function f and draw a line through two points (x, f(x)) and (x + h, f(x + h)) y The slope of the line is: B y2 – y1 x2 – x1 f(x + h) – f(x) (x + h) - x = = f(x + h) – f(x) h A Difference Quotient x x x + h

Difference Quotient Example: Find and simplify the difference quotient for the function f given by f(x) = 3x2 + 2x

Homework PG. 144: 1-4, 12-21(M3), 24, 33, 36, 75, 77, 78 KEY: 24, 36, 78 Reading: 3.2 Quadratic Functions & Graphs