Functions and Graphs
Functions and Graphs
Functions and graphs tests function notation, domain and range, composite and inverse functions, and the drawing and interpretation of straight-line and quadratic graphs - gradients, intercepts, roots, turning points and simple transformations of curves.
Functions
- f(x) = 2x + 1 means substitute: f(3) = 7.
- fg(x) means apply g first, then f. It is not f(x) × g(x).
- Inverse: write y = f(x), swap x and y, then rearrange for y.
- Domain restrictions: a denominator cannot be zero, and a square root needs a non-negative argument.
Straight-line graphs
- y = mx + c: m is the gradient, c the y-intercept.
- Gradient between two points = (y₂ − y₁)/(x₂ − x₁).
- Parallel lines have equal gradients; perpendicular gradients multiply to −1.
Quadratic graphs
- y = ax² + bx + c is a parabola, opening upwards when a > 0.
- Roots where y = 0; axis of symmetry x = −b/2a; the turning point lies on that line.
Transformations
- y = f(x) + a shifts the curve up by a; y = f(x + a) shifts it left by a.
- y = −f(x) reflects in the x-axis; y = kf(x) stretches vertically by factor k.
Worked example. f(x) = 2x − 3. Find f⁻¹(x) and f⁻¹(5). Let y = 2x − 3. Swap: x = 2y − 3. Rearrange: 2y = x + 3, so y = (x + 3)/2. Hence f⁻¹(x) = (x + 3)/2 and f⁻¹(5) = 8/2 = 4.
Traps that cost marks.
- Writing f⁻¹(x) = 1/f(x) - the inverse is not a reciprocal.
- Applying f before g in fg(x).
- Reading the axis of symmetry as x = +b/2a.
- Plotting too few points so the parabola comes out angular rather than smooth.