Functions
The Functions domain tests how well you understand mappings, graphs, domain and range, composition, inverses, and transformations. Higher Level questions often ask you to sketch or interpret a function rather than merely calculate values.
Core rules
- Domain restrictions: a denominator cannot be zero; a square-root radicand must be non-negative; a logarithm requires a positive argument.
- Composite function: ((f\circ g)(x)=f(g(x))). Its domain excludes any (x) not in the domain of (g), and any (x) for which (g(x)) is not in the domain of (f).
- Inverse function: swap (x) and (y) and solve. The domain of (f^{-1}) is the range of (f), and the graphs are reflections in (y=x).
- Transformations: (f(x)+a) shifts vertically; (f(x+a)) shifts horizontally in the opposite direction; (af(x)) stretches vertically; (f(ax)) compresses horizontally by factor (1/a).
- Even function: (f(-x)=f(x)). Odd function: (f(-x)=-f(x)).
- Exponential and log functions are inverses: (y=a^x) and (y=\log_a x), with (e^x) and (\ln x) as the natural pair.
Worked example
Let (f(x)=2x+3) and (g(x)=x^2-1). Then ((f\circ g)(x)=2(x^2-1)+3=2x^2+1). For the inverse, write (y=2x+3), so (x=(y-3)/2), giving (f^{-1}(x)=(x-3)/2). Check: (f(f^{-1}(x))=2\cdot\frac{x-3}{2}+3=x).
Traps
Confusing (f^{-1}(x)) with (1/f(x)); forgetting to restrict the domain of a composite; applying horizontal transformations in the wrong direction; missing asymptotes in sketches; stating a range without considering the graph.