Functions & Graphs
Year 10 (IGCSE) 📊 Algebra Understand function notation, composite and inverse functions.
📐 Function Notation
A function maps each input to exactly one output. We write $f(x)$ to mean "the function $f$ applied to $x$".
Example: If $f(x) = 3x - 1$, then $f(4) = 11$ and $f(-2) = -7$.
⚡ Composite Functions
$$fg(x) = f(g(x)) \quad \text{— apply } g \text{ first, then } f$$Example: $f(x) = x^2$, $g(x) = x+3$. Then $fg(x) = (x+3)^2$ and $gf(x) = x^2 + 3$.
🔄 Inverse Functions
The inverse function $f^{-1}(x)$ reverses the mapping of $f$. If $f$ maps $a \to b$, then $f^{-1}$ maps $b \to a$.
- Write $y = f(x)$.
- Rearrange to make $x$ the subject.
- Replace $x$ with $f^{-1}(x)$ and $y$ with $x$.
Example: $f(x) = 2x + 5$. → $y = 2x+5$ → $x = \frac{y-5}{2}$ → $f^{-1}(x) = \frac{x-5}{2}$
📈 Key Curve Shapes
Recognise the graphs of the most important function families.
| Function | Shape / Key Feature |
|---|---|
| $y = x^2$ | U-shaped parabola, vertex at origin |
| $y = x^3$ | Cubic S-curve through origin |
| $y = \frac{1}{x}$ | Two branches; asymptotes at $x=0$ and $y=0$ |
| $y = a^x$ ($a>1$) | Exponential growth; always positive, passes through $(0,1)$ |
Ready to test yourself? Click the Quiz tab above to answer questions on this topic!
Interactive Demonstration — Functions & Graphs
🟠 f(x) 🔵 g(x) 🟡 gf(x) — composite
📈 Function Grapher
Evaluate f(x), g(x), composite fg(x) and gf(x) at a value.