Operations on Functions
Operations on Functions
Just as you can add, subtract, multiply, and divide numbers, you can perform the same operations on functions. Given two functions f and g, you can form new functions like (f + g)(x), (f − g)(x), (fg)(x), and (f/g)(x) by combining their outputs.
Function composition goes a step further: (f ∘ g)(x) means "first apply g, then apply f to the result." Composition is not commutative — f ∘ g and g ∘ f generally produce different results — and this non-commutativity is an important concept.
This lesson teaches all five operations along with domain considerations for quotients and compositions.
| Operation | Definition |
|---|---|
| Sum | $(f + g)(x) = f(x) + g(x)$ |
| Difference | $(f - g)(x) = f(x) - g(x)$ |
| Product | $(f \cdot g)(x) = f(x) \cdot g(x)$ |
| Quotient | $\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)},\; g(x) \neq 0$ |
$f(x) = 3x + 1$, $g(x) = x - 4$. Find $(f + g)(x)$ and $(f \cdot g)(x)$.
$(f + g)(x) = (3x + 1) + (x - 4) = 4x - 3$.
$(f \cdot g)(x) = (3x + 1)(x - 4) = 3x^2 - 11x - 4$.
Composition of Functions
$(f \circ g)(x) = f(g(x))$. Evaluate $g$ first, then apply $f$ to the result.
$f(x) = 2x + 1$, $g(x) = x^2 - 3$. Find $(f \circ g)(x)$.
$f(g(x)) = f(x^2 - 3) = 2(x^2 - 3) + 1 = 2x^2 - 5$.
Same functions. Find $(g \circ f)(x)$.
$g(f(x)) = g(2x + 1) = (2x + 1)^2 - 3 = 4x^2 + 4x - 2$.
$f \circ g \neq g \circ f$ in general. Composition is not commutative.
$f(x) = 2x + 1$, $g(x) = x^2 - 3$. Find $(f \circ g)(2)$.
$g(2) = 4 - 3 = 1$. Then $f(1) = 2(1) + 1 = 3$.
$f(x) = \sqrt{x}$, $g(x) = x + 4$. Find $(f \circ g)(x)$ and its domain.
$(f \circ g)(x) = \sqrt{x + 4}$. Domain: $x + 4 \ge 0 \Rightarrow x \ge -4$, i.e., $[-4, \infty)$.
Practice Problems
Let $f(x) = 2x - 3$ and $g(x) = x^2 + 1$.
Show Answer Key
1. $x^2 + 2x - 2$
2. $-x^2 + 2x - 4$
3. $f(2) \cdot g(2) = 1 \cdot 5 = 5$
4. $\dfrac{x^2+1}{2x-3}$, domain: $x \neq \dfrac{3}{2}$
5. $2(x^2+1) - 3 = 2x^2 - 1$
6. $(2x-3)^2 + 1 = 4x^2 - 12x + 10$
7. $f(2) = 1$; so $(f \circ g)(1) = f(2) = 1$
8. $f(0) = -3$; $g(-3) = 10$
9. $f(2x - 3) = 2(2x-3) - 3 = 4x - 9$
10. $\sqrt{2x-6}$; $2x - 6 \ge 0 \Rightarrow x \ge 3$, domain $[3,\infty)$