Operations on Functions 6.1 Operations on Functions

Example 1 A. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (f + g)(x).

Example 1 B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find (g– f)(x).

Example 2 A. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find (f ● g)(x).

Remember – you cannot have a 0 in the denominator of a function! Given f(x) = 3x2 – 2x – 1 and g(x) = x2 – 1, find

Example 3 Given f(x) = 2x2 – 3x + 1 and g(x) = x – 1 , find 𝒈 𝒇 (x).

Composite Functions Another way to combine functions is a composition of functions. In a composition of functions, the results of one function are used to evaluate a second function.

Example 4 If f(x) = 2x and g(x) = x2 – 3x + 2 and h(x) = -3x – 4 then find each value. f[g(3)] b. g[h(-2)]

Example 5 If f(x) = (2, 6), (9, 4), (7, 7), (0, –1) and g(x) = (7, 0), (–1, 7), (4, 9), (8, 2), a) Find [f ○ g](x). b) Find [g ○ f](x).

WARM UP If f(x) = {(1, 2), (0, –3), (6, 5), (2, 1)} and g(x) = {(2, 0), (–3, 6), (1, 0), (6, 7)}, find a. f ○ g = b. g ○ f =

Example 6 Find [f ○ g](x) for f(x) = 3x2 – x + 4 and g(x) = 2x – 1. State the domain and range for each combined function.

Example 7 Find [g ○ f](x) for f(x) = 3x2 – x + 4 and g(x) = -2x – 1. State the domain and range for each combined function.

Example 8 Find [f ○ g](x) for h(x) = x2 + 2x + 3 , f(x) = 𝟏 8x + 9 and g(x) = x + 5. State the domain and range for each combined function.

Example 9 If f(x) = axm and g(x) = bxn, perform the operation stated.   A) f(x)  g(x) B) f(g(x)) C) f(g(f(x)))