Functions, Relations and Graphs
What you must understand
A relation is any set of ordered pairs; a function is a relation where each x-value maps to exactly one y-value. Candidates must be fluent with domain and restriction conventions (implied maximal domain), function notation, transformations of standard curves (dilations, reflections, translations in the order dilate-reflect-translate), inverse functions including the relationship between graphs reflected in y = x, and composite functions with their restricted domains. You should recognise the shapes and key features of polynomials, hyperbolas, truncus, square root curves, circles, ellipses and exponential and logarithmic graphs, and be able to sketch with intercepts, asymptotes and turning points labelled.
Traps that appear
- Forgetting that a function is not one-to-one unless it passes the horizontal line test, so its inverse is only a function on a restricted domain.
- Mixing up the order of transformations: y = 2f(x−3) is a dilation by factor 2 from the x-axis followed by a translation 3 right, not a translation of 6.
- Stating the range of a restricted function without checking endpoints.
- Losing solutions when squaring both sides of an equation involving a square root; always check against the original domain.
- Confusing the domain of f(g(x)): it is the set of x in the domain of g with g(x) in the domain of f.
Worked example
Find the inverse of f(x) = x² + 4x for x ≥ −2. Complete the square: f(x) = (x+2)² − 4. On x ≥ −2 the range is y ≥ −4 and the function is one-to-one. Swap and solve: x = (y+2)² − 4 gives y = −2 + √(x+4). So f⁻¹(x) = −2 + √(x+4), domain x ≥ −4. Rule of thumb: always state domain and range alongside the rule.