Function Operations — Definition, Formula & Examples

Function Operations

Definitions for combining functions by adding, subtracting, multiplying, dividing, and composing them.

1. addition $(f+g)(x) = f(x) + g(x)$
2. subtraction $(f-g)(x) = f(x) - g(x)$
3. multiplication $(fg)(x) = f(x) \cdot g(x)$
4. division $\displaystyle \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ ( $g(x) \ne 0\text{)}$
5. composition $(f \circ g)(x) = f(g(x))$

Key Formula

(f + g)(x) = f(x) + g(x) \quad (f - g)(x) = f(x) - g(x)

(f \cdot g)(x) = f(x) \cdot g(x) \quad \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\; g(x) \neq 0

(f \circ g)(x) = f(g(x))

Where:

$f(x)$ = The first function
$g(x)$ = The second function
$f \circ g$ = The composition of f and g, meaning f applied to the output of g

Worked Example

Problem: Given f(x) = 2x + 3 and g(x) = x², find (f + g)(2), (f · g)(2), and (f ∘ g)(2).

Step 1: Evaluate each function at x = 2.

f(2) = 2(2) + 3 = 7, \quad g(2) = 2^2 = 4

Step 2: Add the results for (f + g)(2).

(f + g)(2) = 7 + 4 = 11

Step 3: Multiply the results for (f · g)(2).

(f \cdot g)(2) = 7 \cdot 4 = 28

Step 4: For composition, first apply g, then apply f to that output.

(f \circ g)(2) = f(g(2)) = f(4) = 2(4) + 3 = 11

Answer: (f + g)(2) = 11, (f · g)(2) = 28, and (f ∘ g)(2) = 11.

Why It Matters

Function operations let you build complex models from simpler pieces. For instance, a profit function is the difference of a revenue function and a cost function. Composition is especially important because it describes chained processes, such as applying a tax after a discount.

Common Mistakes

Mistake: Confusing (f ∘ g)(x) with (f · g)(x), or assuming f(g(x)) equals g(f(x)).

Correction: Composition means substituting one function into the other, not multiplying. The order matters: f(g(x)) and g(f(x)) usually give different results.

Related Terms

Composition — Applying one function to the output of another
Function — The basic object being combined
Domain — Must be considered when dividing or composing
Inverse Function — Composition with its inverse yields the identity