What are operations on functions?

Just like numbers, functions can be combined using operations: addition, subtraction, multiplication, division, and composition.

If we have two functions:

We can create new functions using these operations.

1️⃣ Addition of functions

Rule:

$$(f+g)(x)=f(x)+g(x)$$

• Add the outputs of the functions for each input.

Example:

$$(f+g)(x)=(2x+3)+(x-1)=3x+2$$

✅ New function: (f + g)(x) = 3x + 2

2️⃣ Subtraction of functions

Rule:

$$(f-g)(x)=f(x)-g(x)$$

Example:

$$(f-g)(x)=(5x+4)-(2x-1)=3x+5$$

✅ New function: (f - g)(x) = 3x + 5

3️⃣ Multiplication of functions

Rule:

$$(f\cdot g)(x)=f(x)\cdot g(x)$$

• Multiply the outputs for each input.

Example:

(f⋅g)(x)=(x+2)(x−3)=x2−x−6(f \cdot g)(x) = (x + 2)(x – 3) = x^2 – x – 6(f⋅g)(x)=(x+2)(x−3)=x2−x−6

✅ New function: (f · g)(x) = x^2 - x - 6

4️⃣ Division of functions

Rule:

(fg)(x)=f(x)g(x),g(x)≠0\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad g(x) \neq 0(gf​)(x)=g(x)f(x)​,g(x)=0

• Divide the outputs for each input.

• Remember: cannot divide by zero.

Example:

(fg)(x)=x2−1x−1=(x−1)(x+1)x−1=x+1,x≠1\left(\frac{f}{g}\right)(x) = \frac{x^2 – 1}{x – 1} = \frac{(x – 1)(x + 1)}{x – 1} = x + 1, \quad x \neq 1(gf​)(x)=x−1x2−1​=x−1(x−1)(x+1)​=x+1,x=1

✅ New function: (f / g)(x) = x + 1, x ≠ 1

5️⃣ Composition of functions

Rule:

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

• Substitute g(x) into f(x).

• Think of it as “g goes first, then f”.

Example:

$$(f\circ g)(x)=f(g(x))=f(x-3)=2(x-3)+1=2x-5$$

✅ New function: (f ∘ g)(x) = 2x - 5

Tip: Composition is not commutative: (f ∘ g)(x) ≠ (g ∘ f)(x) in most cases.

Why operations on functions matter

• Allows you to combine multiple relationships

• Used in real-world modeling: combining costs, distances, or transformations

• Foundation for advanced algebra and calculus

Common beginner mistakes

1. ❌ Forgetting parentheses when substituting

2. ❌ Not simplifying correctly

3. ❌ Ignoring domain restrictions in division

4. ❌ Mixing up (f + g)(x) with f(x) + g (the function vs. its value)

Real-life example

• f(x) = cost of ingredients for x cakes

• g(x) = cost of labor for x cakes

• Total cost: (f + g)(x)

• Profit: (Revenue - Cost)(x)

• Special discounts applied: (f ∘ g)(x)

Operations on functions let you model complex real-world situations algebraically.