📌 Purpose: Understand what a function is, how to express it, determine domain and range, interpret graphs, and find/check inverse functions. Clear definitions, short examples, and practical tips for exams and calculators.
| Term | Definition / Note |
|---|---|
| Function | A rule f that assigns to each input x in the domain exactly one output f(x). Notation: f(x), v(t), C(n) etc. |
| Domain | Set of permitted inputs (x-values) for which f(x) is defined. Example: for f(x)=√(2 − x), domain x ≤ 2. |
| Range | Set of possible outputs f(x). Example above: range f(x) ≥ 0. |
| Graph of a function | Visual representation: points (x, f(x)). Useful to see continuity, domain/range, intercepts and inverse reflection across y = x. |
| One-to-one (injective) | Each y in range is image of exactly one x. Horizontal line test: any horizontal line intersects graph at ≤ 1 point. |
| Inverse function f−1 | Function that reverses f: f−1(y) = x iff f(x) = y. Exists only for one-to-one functions; domain of f−1 = range of f. |
📌 Determining domain & range (quick rules)
- Polynomials: domain all real numbers (ℝ). Range depends on degree and turning points.
- Rational functions: exclude x where denominator = 0 (vertical asymptotes).
- Roots: for even roots (√), require expression inside ≥ 0 (or solve inequality to find domain).
- Logarithms: argument must be > 0.
- Graphical approach: domain = projection of graph on x-axis; range = projection on y-axis.
📌 Inverse functions: meaning & how to find
- Meaning: f−1 undoes f: if y = f(x) then x = f−1(y). Graphically, reflect graph of f across line y = x.
- Algebraic method (standard):
- Write y = f(x).
- Swap x and y: x = f(y).
- Solve for y in terms of x; result is f−1(x).
- Existence: Only invertible if one-to-one. If not one-to-one, restrict domain to a region where it is one-to-one (common for sqrt, sin, etc.).
- Check: Verify f(f−1(x)) = x and f−1(f(x)) = x on valid domains.
📌 Short examples
Example 1 (domain & range): f(x) = √(2 − x). Domain: x ≤ 2. Range: f(x) ≥ 0.
Example 2 (finding inverse): f(x) = 2x + 3. Let y = 2x + 3. Swap: x = 2y + 3 ⇒ y = (x − 3)/2. So f−1(x) = (x − 3)/2. Check: f(f−1(x)) = x.
Example 3 (non-invertible without restriction): f(x) = x2 is not one-to-one on ℝ (fails horizontal line test); restrict to x ≥ 0 to get inverse f−1(x) = √x.
🧮 GDC Tips
- Plot f(x) to inspect domain/range visually and to test one-to-one property (horizontal line test).
- Use solver or algebra tools to swap x & y and solve for inverse; if GDC gives numeric inverse, also show algebraic steps in exam work.
- For inverse check numerically: compute f(f−1(a)) and f−1(f(a)) for sample a in domain to validate.
🧠 Examiner Tip
When asked for domain/range show brief reasoning (e.g., set inside √ ≥ 0 or denominator ≠ 0). For inverses: explicitly state domain restriction if needed and demonstrate both compositions f(f−1(x)) and f−1(f(x)).
📝 Paper tips
- State domain explicitly and show algebraic steps (not just final interval).
- In context questions (temperature/currency), explain units and meaning of inverse (e.g., converting back).
⚠️ Common pitfalls
- Forgetting to restrict domain for inverse when function is not one-to-one.
- Confusing domain and range — remember domain = input x-values, range = outputs f(x).
- Not checking compositions to confirm inverse.
📌 Quick checklist
- Identify expression type (polynomial, rational, root, log) → apply domain rules.
- Find range by algebra or by analyzing graph/transformations.
- To find inverse: swap x & y, solve for y, then restrict domain if necessary.
- Always check inverses via composition f(f−1(x)) and f−1(f(x)).