Relations, Functions and Graphs
Relations, Functions and Graphs
Candidates must distinguish relations from functions, state domain and range, use function notation, and interpret graphs. Know the vertical line test, slope and intercepts, linear and quadratic forms, transformations, and basic exponential behaviour. Be able to find inverses, evaluate composite functions, and read intervals of increase or decrease from a graph.
Traps: treating a vertical line as a function; confusing f(x + h) with f(x) + h; forgetting domain restrictions such as a denominator being nonzero or a square root requiring a nonnegative radicand; and reversing the slope sign when reading a graph. For inverses, swap x and y and solve, then check that the inverse is a function. Horizontal lines have slope 0; vertical lines have undefined slope. A parabola's vertex is a maximum or minimum, not necessarily an x-intercept.
Rule of thumb: use the vertical line test for functions and the horizontal line test for one-to-one functions. Slope is rise over run, and for y = mx + c the gradient is m and the y-intercept is c.
Worked example: find the inverse of f(x) = 2x + 3. Write y = 2x + 3, swap x and y: x = 2y + 3, so y = (x − 3)/2. Thus f⁻¹(x) = (x − 3)/2. Check f(f⁻¹(x)) = x.