Functions
Core skills
Functions questions test the idea of a rule that assigns exactly one output to each input. You need function notation, domain and range, composition, inverses, transformations of graphs, and the behavior of the main families: linear, quadratic, polynomial, exponential, rational, radical, and logarithmic. Piecewise definitions, average rate of change over an interval, and arithmetic/geometric sequences also appear.
Traps to expect
- Inside versus outside changes. $f(x + 2)$ shifts the graph left 2; $f(x) + 2$ shifts it up 2. Multiplying outside ($2f(x)$) stretches vertically; multiplying inside ($f(2x)$) compresses horizontally, and reflections follow the same inside/outside logic.
- Composition order matters. $f(g(x))$ means substitute $g(x)$ into $f$, working from the inside out.
- Domain restrictions: exclude values that make a denominator zero or put a negative under an even root.
- Inverses swap roles. To find $f^{-1}(a)$, solve $f(x) = a$ for $x$ - no need to derive the general inverse.
- Exponential versus linear growth: constant differences mean linear, constant ratios mean exponential.
Worked example
If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, find $f(g(2))$. Work inside out: $g(2) = 4 - 1 = 3$, then $f(3) = 2(3) + 3 = 9$. Reversing the order gives $g(f(2)) = g(7) = 48$ - a very different answer, which is why the order matters.
Rule of thumb
When a graph transformation question appears, track a single point. Ask where the point $(0,0)$ or $(1,1)$ of the parent graph ends up; that identifies the transformation faster than reasoning about the whole shape.