Functions and Graphs
Functions and Graphs
This domain checks whether you can treat a function as an object - its domain, range and behaviour - and transform it. MATSEC questions typically ask for a composite, an inverse, a sketch showing asymptotes, or a solution read from a graph. Precision in stating domains is where most marks are won or lost.
Core rules
- fg(x) means apply g first, then f; the domain of fg excludes any x not in the domain of g, and any x for which g(x) lies outside the domain of f.
- An inverse exists only for a one-to-one function; to find it, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f.
- Transformations: y = f(x) + a shifts up by a; y = f(x + a) shifts left by a; y = af(x) stretches vertically by factor a; y = f(ax) compresses horizontally by factor 1/a.
- |f(x)| reflects any part below the x-axis upwards; f(|x|) reflects the right-hand branch onto the left.
Worked example
f(x) = (x + 1)/(x − 2), x ≠ 2. Find f⁻¹. Write y = (x + 1)/(x − 2). Multiply: y(x − 2) = x + 1, so yx − 2y = x + 1 and x(y − 1) = 1 + 2y. Hence x = (2y + 1)/(y − 1), so f⁻¹(x) = (2x + 1)/(x − 1), x ≠ 1. The excluded value 1 is exactly the value f can never take, confirming that domain and range swap.
Traps
- Applying f before g in a composite.
- Stating an inverse without excluding the value that makes its denominator zero.
- Confusing f(x + 2) with f(x) + 2 - the first moves left, the second up.
- Giving a range as an interval without checking whether the endpoints are attained.