Combining functions
Functions can be added, subtracted, multiplied and divided, just like numbers. Combine their outputs at each x.
| Operation | Notation | Rule |
|---|---|---|
| Sum of functions | (f + g)(x) | f(x) + g(x) |
| Difference of functions | (f − g)(x) | f(x) − g(x) |
| Product of functions | (f · g)(x) | f(x) · g(x) |
| Quotient of functions | (f/g)(x) | f(x)/g(x), g(x) ≠ 0 |
Example
f(x) = x² − 1, g(x) = x + 1
- (f + g)(x) = x² + x
- (f − g)(x) = x² − x − 2
- (f · g)(x) = (x² − 1)(x + 1) = x³ + x² − x − 1
- (f/g)(x) = (x² − 1)/(x + 1) = x − 1, x ≠ −1
Evaluating at a number
With the same f and g:
- (f + g)(3) = 8 + 4 = 12
- (f · g)(2) = 3 × 3 = 9
Domain
The domain of a combination is where both functions are defined. For quotients, also exclude zeros of the denominator.
Example: f(x) = √x, g(x) = x − 2
- (f + g): domain x ≥ 0
- (f/g)(x) = √x/(x − 2): domain x ≥ 0, x ≠ 2
Adding graphs
To graph f + g, add the y-values at each x. This is called superposition.
| x | f(x) | g(x) | (f + g)(x) |
|---|---|---|---|
| 0 | 1 | 2 | 3 |
| 1 | 3 | 1 | 4 |
| 2 | 5 | 0 | 5 |
Real-world combinations
Profit = revenue − cost
- R(x) = 25x (price × quantity)
- C(x) = 8x + 1,200 (variable + fixed costs)
- P(x) = R(x) − C(x) = 17x − 1,200
Break-even when P(x) = 0: x ≈ 70.6, so 71 units.
Total sound or light: waves add by superposition.
Average cost: A(x) = C(x)/x, a quotient of functions.
Not the same as composition
(f · g)(x) multiplies outputs; f(g(x)) feeds one into the other. They’re different!
f(x) = x + 1, g(x) = 2x:
- (f · g)(x) = (x + 1)(2x) = 2x² + 2x
- f(g(x)) = 2x + 1
Worked examples
Example 1: f(x) = 3x, g(x) = x². (f − g)(4)? 12 − 16 = −4.
Example 2: f(x) = x + 2, g(x) = x − 2. (f · g)(x)? x² − 4.
Example 3: Domain of (f/g)(x) with f(x) = x and g(x) = x² − 9? x ≠ ±3.
Common mistakes
- Confusing product with composition.
- Forgetting brackets when subtracting: f − g = f(x) − (x + 1).
- Missing domain restrictions for quotients.
Try it yourself
f(x) = 2x + 1, g(x) = x − 3
- (f + g)(x)
- (f − g)(x)
- (f · g)(x)
- (f/g)(x) and its domain
- (f + g)(5)
Answers: 1) 3x − 2 2) x + 4 3) 2x² − 5x − 3 4) (2x + 1)/(x − 3), x ≠ 3 5) 13
Quick recap
- Combine functions by adding, subtracting, multiplying or dividing their outputs.
- The domain is where both are defined (and the denominator ≠ 0).
- Profit = revenue − cost is a difference of functions.