1. Ch. 1 – Functions and Their Graphs
1.5 – Combinations of Functions

2. Arithmetic Combinations
Functions can be added, subtracted, multiplied, or divided to produce new functions.
Ex: Given f(x) = 2x – 3 and g(x) = x2 +2x – 5, find:
(f + g)(x)
Just add the two functions!
2x – 3 + x2 + 2x – 5 = x2 + 4x – 8
(g – f)(x)
Just subtract the two functions!
x2 + 2x – 5 – (2x – 3) = x2 – 2
Don’t multiply the number or variable in ( ) by the function!!!

3. Find (fg)(x). x2 + 3x – 18 x2 – 21x + 45 x3 – 3x2 + 2x – 15

4. Find (f/g)(x).
x + 5
x2 – 3
x – 3

5. What is the domain of (f/g)(x)?
(-∞, 0) and (0, ∞)
(-∞, 3) and (3, ∞)
(-∞, -5] and [3, ∞)
(-∞, -5) and (3, ∞)

6. Composition of Functions
For 2 functions f(x) and g(x), a composition of f with g is f(g(x)) or and is read as “f of g”
Ex: Given f(x) = 2x – 3 and g(x) = x2 +2x – 5, find
Put g(x) in for x in f(x)!
f(g(x)) = f(x2 + 2x – 5)
= 2(x2 + 2x – 5) – 3
= 2x2 + 4x – 10 – 3
= 2x2 + 4x - 13

7. Find x

8. Domain of Compositions
To find the domain of , first find the domain of f, then restrict g to that domain
Ex: Given and g(x) = 2x, find the domain of
First, find the domain of f…
4 –x ≥ 0
4 ≥ x
x ≤ 4
Now restrict g to that domain…
2x ≤ 4
x ≤ 2!

9. Find the domain of . xϵℝ (-∞, 0] and [0, ∞) (-∞, -15] and [-15, ∞)
[-5, ∞)
(-∞, -5] and [-5, ∞)