A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Functions
CSCA Mathematics · Lesson 4 of 16Functions and graphs
Fonctions et graphes · الدوال ومنحنياتها
▶ Watch the video lesson (A7LAM students)
A function links each input x to exactly one output f(x). The test asks you to evaluate, compose and read graphs of functions.
Key ideas
- The domain is the set of x values where the function is defined (no division by 0, no square root of a negative).
- A linear function f(x) = mx + c has slope m and y-intercept c.
- A parabola f(x) = ax² + bx + c has its vertex at x = −b / (2a). It opens up if a > 0.
- Composition: (f∘g)(x) = f(g(x)). The inverse f⁻¹ undoes f.
Formulas to know
- Slope m = (y₂ − y₁) / (x₂ − x₁)
- Vertex x = −b / (2a)
- If y = f(x), solve for x to find f⁻¹
Worked example
f(x) = 2x + 1 and g(x) = x². Find f(g(3)) and g(f(3)).
- g(3) = 9, so f(g(3)) = f(9) = 2·9 + 1 = 19.
- f(3) = 7, so g(f(3)) = g(7) = 49.
- Order matters: f∘g is not g∘f.
Answer: f(g(3)) = 19 and g(f(3)) = 49
Practice questions
1. What is the slope of the line through (1, 2) and (3, 8)?
2 · 3 ✓ · 4 · 6
m = (8 − 2) / (3 − 1) = 6 / 2 = 3.
2. What is the domain of f(x) = √(x − 4)?
x > 4 · x ≥ 4 ✓ · x ≤ 4 · All real numbers
We need x − 4 ≥ 0, so x ≥ 4.
3. Find the inverse of f(x) = 3x − 6.
f⁻¹(x) = (x + 6)/3 ✓ · f⁻¹(x) = 3x + 6 · f⁻¹(x) = x/3 − 6 · f⁻¹(x) = (x − 6)/3
y = 3x − 6 gives x = (y + 6)/3, so f⁻¹(x) = (x + 6)/3.