Functions: notation, domain and range, composite and inverse functions, sketching graphs

Notation & Basic Idea

Think of a function as a machine that takes an input \(x\) and gives an output \(y\). We write it as \(y = f(x)\) or simply \(f(x)\) when the output is understood.

• Domain – all possible inputs.
• Range – all possible outputs.

Example: \(f(x)=x^2\) takes any real number, but only non‑negative numbers appear as outputs.

Domain & Range – How to Find Them

🔍 Domain: Look for restrictions like division by zero or square roots of negative numbers.

📈 Range: Find the minimum/maximum values or use a graph.

Composite Functions – Putting Machines Together

When you feed the output of one function into another, you get a composite function:

\( (g\circ f)(x)=g(f(x)) \)

1. Compute \(f(x)\).
2. Plug that result into \(g\).

Example: Let \(f(x)=x+2\) and \(g(x)=x^2\). Then \( (g\circ f)(x)= (x+2)^2 \).

⚠️ Domain of a composite is all \(x\) that are in the domain of \(f\) and for which \(f(x)\) lies in the domain of \(g\).

Inverse Functions – Reversing the Machine

An inverse function \(f^{-1}\) undoes what \(f\) does: \(f^{-1}(f(x))=x\) and \(f(f^{-1}(y))=y\).

To find it:

1. Write \(y=f(x)\).
2. Swap \(x\) and \(y\).
3. Solve for the new \(y\).

Example: For \(f(x)=3x+5\), swap to get \(x=3y+5\), then \(y=\dfrac{x-5}{3}\). So \(f^{-1}(x)=\dfrac{x-5}{3}\).

🔑 Check by plugging back in.

Sketching Graphs – Visualising Functions

Use these steps:

1. Plot key points (e.g., intercepts, turning points).
2. Mark the domain limits.
3. Sketch the curve, keeping in mind symmetry (even/odd).
4. Check for asymptotes if the function involves division.

Example: Sketch \(y=x^2-4\).

• Vertex at \( (0,-4) \).
• X‑intercepts at \(x=\pm2\).
• Y‑intercept at \(y=-4\).

Draw a U‑shaped curve passing through these points.

📌 Examination Tips

• Always state the domain and range when asked.
• When finding an inverse, check by composition to avoid mistakes.
• For composite functions, remember to check the domain of the inner function first.
• Use a table of values to help sketch graphs accurately.
• Practice identifying asymptotes for rational functions.

Good luck! 🎯