Functions

2.12 Functions – Cheat Sheet

1. Functions, Domain, and Range

• Function: A rule mapping each input (x) to exactly one output (y).
• Notation: f(x) means “function f of x”.
• Domain: All allowed input values for x.
• Range: All possible output values for y.

Example: f(x) = 3x − 5

• Domain: all real numbers (unless restricted).
• Range: all real numbers.

Mapping Diagram: Shows how each element in the domain maps to exactly one element in the range.

2. Function Notation Examples

3. Inverse Functions (f⁻¹(x))

• Reverses the effect of the function.
• Steps to find inverse:
  1. Write y = f(x).
  2. Swap x and y.
  3. Rearrange to make y the subject.
  4. Replace y with f⁻¹(x).

Example: f(x) = 3x − 5

1. y = 3x − 5
2. Swap: x = 3y − 5
3. x + 5 = 3y → y = (x + 5)/3
4. f⁻¹(x) = (x + 5)/3

4. Composite Functions (gf)(x) = g(f(x))

• Apply f first, then g to the result.

Example 1:
f(x) = 3x − 5, g(x) = (x + 4)/5

• f(x) = 3x − 5
• g(f(x)) = [(3x − 5) + 4] / 5 = (3x − 1)/5

Example 2:
f(x) = 3/(x + 2), g(x) = (3x + 5)²

• f(x) = 3/(x + 2)
• g(f(x)) = [3(3/(x + 2)) + 5]² = [(9/(x + 2)) + 5]²
• Simplify: [(9 + 5(x + 2)) / (x + 2)]² = [(9 + 5x + 10) / (x + 2)]² = [(5x + 19) / (x + 2)]²

5. Quick Function Rules Table

6. Domain Restrictions

• Cannot divide by zero → Denominator ≠ 0.
• Even root (√) requires non-negative inside.

Example: f(x) = 1/(x − 3) → Domain: x ≠ 3

7. Examples Table for Quick Reference