Functions
What Functions tests
Functions is about relationships between variables: domain and range, composition, inverses, transformations, and standard families such as quadratics, exponentials and rational functions. The exam often asks you to justify a domain restriction or read a graph.
Core rules
- Domain: exclude values making a denominator zero; require a radicand under an even root to be non-negative; require a log argument positive.
- Composite (f∘g)(x) = f(g(x)): its domain contains x in the domain of g for which g(x) is in the domain of f.
- Inverse: swap x and y and solve; f⁻¹ exists only if f is one-to-one. Its graph is the reflection of y = f(x) in y = x; domain and range swap.
- Transformations: y = af(b(x − h)) + k has vertical stretch |a|, horizontal stretch 1/|b|, translation h right and k up; negative a or b reflects.
- Quadratic y = a(x − h)^2 + k has vertex (h,k); discriminant b^2 − 4ac decides the number of real roots.
Worked example
Let f(x) = 2(x − 3)^2 + 1 for x ≥ 3. Find f⁻¹ and state its domain. Set y = 2(x − 3)^2 + 1. Then y − 1 = 2(x − 3)^2, so x − 3 = sqrt((y − 1)/2) because x ≥ 3. Thus f⁻¹(x) = 3 + sqrt((x − 1)/2). The range of f is y ≥ 1, so the domain of f⁻¹ is x ≥ 1; its range is y ≥ 3.
Traps
Writing f⁻¹ as 1/f; inverting a quadratic without restricting its domain; forgetting to swap domain and range; taking the negative square root when the restriction requires the positive one; applying horizontal transformations in the wrong order.