Pre-Calc Lesson 4.2 Operations on Functions We can do some basic operations on functions. We can add, subtract, multiply and divide functions. If given functions f(x) and g(x), then: Sum of functions f and g : (f + g)(x) = f(x) + g(x) Difference of functions f and g: (f - g)(x) = f(x) – g(x) Product of functions f and g: (f g)(x) = f(x) g(x) Quotient of functions f and g: (f/g)(x) = f(x)/g(x) where g(x) ≠ 0

Let f(x) = x + 1 and g(x) = x2 – 1, find a ‘rule’ for each of Example 1: Let f(x) = x + 1 and g(x) = x2 – 1, find a ‘rule’ for each of the following functions. (f + g)(x)  think this means f(x) + g(x)  (x+1) + (x2 – 1) (drop all parentheses)  x + 1 + x2 - 1 (rearrange terms)  x2 + x (voila!) (f/g)(x) think  f(x) g(x) replace  x + 1 x2 – 1 reduce if possible ??  (x + 1) factor  (x+1)(x-1) Cancel like factors  1 (x – 1) ta da!  but we must state: x ≠ + 1 know why ?????

Another way of combining functions is called: Composition of functions! This is simply a process of substituting a functions ‘rule’ In for the variable in a 2nd function. Example 2: Let f(x) = x4 – 3x2 and g(x) = √(x – 2) Find the composition of functions f and g. This means  (f of g)(x) this is what it looks like f(g(x)) and this means  Substitute the rule from g(x) in to the variable in the rule of f(x) ??? i.e. Plug √(x – 2) in for ‘x’ in the rule (x)4 - 3(x)2  (√(x – 2))4 – 3(√(x – 2))2  ( √(x – 2)2)2 - 3(x – 2)  (x – 2)2 – 3x + 6  x2 – 4x + 4 – 3x + 6  x2 – 7x + 10 for which x > 2 ?? Voila!

Let f(x) = 1/x and g(x) = x + 1, find new ‘rules’ for f(g(x)) Example 3: Let f(x) = 1/x and g(x) = x + 1, find new ‘rules’ for f(g(x)) and g(f(x)) and give the domain of each new ‘composite’ function. f(g(x)) = f(x + 1) 1 (x + 1) ; x ≠ - 1 g(f(x)) = g( 1 ) x ( 1 ) + 1 and x ≠ 0 Hw: pgs 127-129: CE: #1-13 all, WE: #1-19 odd