Functions and Graphs
Core concepts
Understand domain and range, function notation, composition, inverses, transformations, and the standard families (linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric). Know how shifts, stretches, and reflections change a graph, and how to read intercepts, asymptotes, and symmetry.
Traps
- Composition order: f(g(x)) ≠ g(f(x)) in general.
- Inverse functions require one-to-one; restrict the domain if needed (e.g., square root for x²).
- Horizontal vs vertical transformations: f(x − h) shifts right by h; f(kx) compresses horizontally by 1/k.
- Asymptotes: a rational function's vertical asymptotes come from zeros of the denominator after cancellation; horizontal asymptotes depend on degree comparison.
- Domain of composition: must also respect the inner function's domain.
Rule of thumb
To find an inverse, swap x and y in y = f(x), then solve for y; verify f(f⁻¹(x)) = x on the restricted domain.
Worked example
Let f(x) = 2x + 3. Find f⁻¹. Write y = 2x + 3; swap to x = 2y + 3; solve y = (x − 3)/2. Check: f(f⁻¹(x)) = 2·(x − 3)/2 + 3 = x. The inverse is f⁻¹(x) = (x − 3)/2.