Functions and Graphs
Core concepts
Function questions test domain and range, composition, inverses, transformations, and the shapes of standard graphs: linear, quadratic, polynomial, exponential, logarithmic, rational, and absolute value. You should be able to shift, stretch, and reflect a graph from its equation, read intercepts and asymptotes, and determine whether a relation is a function (vertical line test) or invertible (horizontal line test). Understand that f(g(x)) is not generally g(f(x)), and that the inverse undoes the original only on a restricted domain where the function is one-to-one.
Traps to watch
- f(x + 2) shifts the graph left by 2, not right.
- 2f(x) stretches vertically; f(2x) compresses horizontally by a factor of 1/2.
- The domain of a composite function excludes inputs where the inner function is undefined.
- Exponential growth outpaces any polynomial eventually; logarithms grow slowest.
- Asymptotes are not always intercepts - a rational function may cross a horizontal asymptote.
Rule of thumb
For transformations, apply horizontal changes inside the parentheses in reverse order and with reversed sign; apply vertical changes outside in the order written.
Worked example
Let f(x) = sqrt(x - 3) and g(x) = x^2 + 1. Find the domain of f(g(x)). We need g(x) - 3 ≥ 0, so x^2 - 2 ≥ 0, giving x ≤ -sqrt(2) or x ≥ sqrt(2). Note that the domain of g alone is all reals; the restriction comes entirely from f.