Recognise and use function notation such as f(x), f⁻¹(x), fg(x) and f²(x)

🔢 Function Notation

In maths a function assigns each input value $x$ to exactly one output value. We write this as $f(x)$.

• Think of a vending machine: you press a button (input) and it gives you a snack (output).
• Example: $f(x)=2x+1$ means if you give the machine the number 3, it returns 7.

🔄 Inverse Functions (f⁻¹(x))

An inverse function $f^{-1}(x)$ reverses the action of $f(x)$. If $f(a)=b$ then $f^{-1}(b)=a$.

1. Draw a table of $x$ and $f(x)$.
2. Swap the columns to get $f^{-1}(x)$.

Example: If $f(x)=3x-2$, then $f^{-1}(x)=\frac{x+2}{3}$.

🔗 Composition of Functions (f∘g)(x) = f(g(x))

Composition means you first apply $g$ then $f$. Think of a two‑step recipe.

Example: Let $f(x)=x^2$ and $g(x)=x+1$. Then $(f∘g)(x)= (x+1)^2$.

🌀 Powers of Functions (f²(x) = f(f(x)))

Applying a function to itself. It’s like turning a page twice in a book.

• Example: If $f(x)=x+3$, then $f^2(x)=f(f(x))=x+6$.
• For quadratic functions, $f^2(x)$ can get large quickly.

🏆 Examination Tips

1. Identify the function first: Write down $f(x)$ clearly before doing any operations.
2. Check domain & range: Inverse functions only exist if the original is one‑to‑one.
3. Use tables for inverses: It’s quick and reduces algebraic errors.
4. Remember composition order: $f∘g$ is not the same as $g∘f$ unless both commute.
5. Practice power notation: Write $f^2(x)$ as $f(f(x))$ to avoid confusion.