Functions and Graphs
Functions and graphs examines how relationships are represented, transformed and interpreted. Questions cover function notation, composite and inverse functions, straight-line and quadratic graphs, and recognition of cubic, reciprocal, exponential and trigonometric curves.
Core rules
- f(a) means substitute a for x; fg(x) means f(g(x)), worked from the inside out.
- Inverse: write y = f(x), swap x and y, then rearrange for y. The domain of f⁻¹ is the range of f.
- y = mx + c has gradient m and y-intercept c.
- y = ax² + bx + c is a parabola; the axis of symmetry is x = −b ÷ 2a, and the turning point follows from completing the square or substituting that x.
- Transformations: y = f(x) + a shifts up by a; y = f(x + a) shifts left by a; y = −f(x) reflects in the x-axis; y = kf(x) stretches vertically by factor k.
- Solutions of f(x) = g(x) are the x-coordinates of the intersection points of the two graphs.
Worked example
Given f(x) = 2x − 3 and g(x) = x² + 1, find fg(2) and f⁻¹(x). Reasoning: g(2) = 2² + 1 = 5, then f(5) = 2(5) − 3 = 7, so fg(2) = 7. For the inverse, let y = 2x − 3; swapping gives x = 2y − 3, so y = (x + 3)/2 and f⁻¹(x) = (x + 3)/2. Check: f⁻¹(7) = 5, returning the input that produced 7.
Traps that cost marks
- Treating fg(x) and gf(x) as the same function.
- Shifting in the wrong direction for f(x + a).
- Writing the inverse as 1/f(x).
- Forgetting that x² = 9 has two roots unless the domain excludes negatives.
- Reading a gradient from a graph with inconsistent square counting.