Find the domain and range of functions, including inverse and composite functions

📐 Functions – Domain, Range, Inverse & Composite

Welcome to your quick‑guide to functions for the Cambridge IGCSE Additional Mathematics 0606 exam. Think of a function as a machine that takes an input (the domain) and gives you an output (the range). We’ll explore how to find these, and how to reverse or combine machines.

🔍 1. Domain – What can you put in?

The domain is the set of all real numbers that can be used as inputs without breaking the function.

• For polynomials like $f(x)=x^3-4x$, every real number works.
• For rational functions, you must avoid values that make the denominator zero. Example: $g(x)=\frac{1}{x-2}$ → domain is all real numbers except $x=2$.
• For square roots, the expression inside the root must be non‑negative. Example: $h(x)=\sqrt{x+5}$ → domain $x\ge -5$.

Analogy: Think of the domain as the “allowed keys” that fit into a lock. If you try a key that doesn’t fit, the lock (function) won’t work.

📈 2. Range – What comes out?

The range is the set of all possible outputs.

1. For $f(x)=x^2$, the range is $y\ge 0$ because squares are never negative.
2. For $g(x)=\frac{1}{x-2}$, the range is all real numbers except $y=0$ (the function never equals zero).
3. For $h(x)=\sqrt{x+5}$, the range is $y\ge 0$ (square roots are non‑negative).

Analogy: If the domain is the keys, the range is the collection of doors you can open. Some doors (outputs) may never be reachable.

🔁 3. Inverse Functions – Turning the machine around

An inverse function, written $f^{-1}(x)$, swaps the roles of input and output.

• To find $f^{-1}(x)$, swap $x$ and $y$ in $y=f(x)$ and solve for $y$.
• Example: $f(x)=2x+3$ → $y=2x+3$ → swap → $x=2y+3$ → $y=\frac{x-3}{2}$ → $f^{-1}(x)=\frac{x-3}{2}$.
• Domain of $f^{-1}$ is the range of $f$, and vice versa.

Exam tip: Always check that $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$ to confirm you’ve got the right inverse.

🧩 4. Composite Functions – Stacking machines

A composite function $f\circ g$ means you first apply $g$, then $f$: $(f\circ g)(x)=f(g(x))$.

1. Domain of $f\circ g$ is all $x$ in the domain of $g$ such that $g(x)$ lies in the domain of $f$.
2. Example: Let $f(x)=x^2$ and $g(x)=x-1$.
   • Domain of $g$ is all real numbers.
   • Domain of $f$ is all real numbers.
   • So domain of $f\circ g$ is all real numbers.
   • $(f\circ g)(x) = (x-1)^2$.

Analogy: Think of $g$ as a “pre‑processor” that prepares data for $f$. The composite is like a two‑stage factory line.

📚 5. Quick Example Table

📝 6. Examination Tips

• Read the question carefully. Identify whether it asks for domain, range, inverse, or composite.
• Check for restrictions. Look for denominators, square roots, logarithms, etc.
• Use a table of values. Plug in a few numbers to spot patterns.
• Remember the inverse rule. Swap $x$ and $y$, solve for $y$, then swap back.
• For composites, follow the order. Find the domain of the inner function first, then ensure the outer function accepts those outputs.
• Use the domain–range swap trick for inverses: the domain of $f^{-1}$ is the range of $f$.
• Mark your work clearly – examiners look for logical steps.

💡 7. Practice Problems

1. Find the domain and range of $p(x)=\frac{3x}{x+1}$.
2. Determine $q^{-1}(x)$ for $q(x)=\frac{2x-5}{x+3}$.
3. Let $r(x)=\sqrt{4-x}$ and $s(x)=x^2-1$. Find the domain of $(r\circ s)(x)$.
4. Sketch the graph of $t(x)=\frac{1}{x-4}$ and state its asymptotes.
5. Given $u(x)=\ln(x-2)$, find its domain and range.

Try solving them before checking the solutions in your textbook. Practice is the best way to master these concepts!