This module covers how functions work in Additional Mathematics, including the domain (valid inputs) and the range (outputs). You combine two functions into a composite and reverse a function with an inverse.
You also see why some functions have no inverse until the domain is restricted. One idea runs through all of it: what a function accepts and what it returns must always be stated, not assumed.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content and notation in your exam year, including any given formulae and calculator rules. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to factorise quadratics, complete the square, rearrange formulae and solve simple quadratic equations. If completing the square is slow, revise quadratic structure and discriminants alongside this module, since it gives you the turning point that decides a range.
An orienting example
The function f is defined by f(x) = x² − 4x + 3 for x ≥ 2. Find the range of f, and then find f⁻¹(x).
Step 1, complete the square: f(x) = (x − 2)² − 1.
Step 2, range: on x ≥ 2 the smallest value is f(2) = −1, and f only increases after that. So the range is f(x) ≥ −1.
Step 3, inverse: y = (x − 2)² − 1, so (x − 2)² = y + 1, and since x ≥ 2 we take the positive root: x = 2 + √(y + 1).
Step 4, state it fully: f⁻¹(x) = 2 + √(x + 1) for x ≥ −1.
Check: f(4) = 16 − 16 + 3 = 3. Then f⁻¹(3) = 2 + √4 = 4. It returns to 4.
That one question used a restricted domain, a range, an inverse and the link between them: the domain of f⁻¹ is the range of f.
In which order should you study it?
- Find valid inputs for a rational expression: the simplest domain idea, where a zero denominator is the only danger.
- Determine a range from a restricted domain: reads outputs from an interval using endpoints and turning points.
- Form a composite function in the correct order: builds fg and gf and shows why order matters.
- Find an inverse and check its domain: rearranges for x and links the inverse’s domain to the range.
- Explain a many-to-one mapping that has no unrestricted inverse: says why an inverse can fail and how a restriction repairs it.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Cancelling before stating the restriction, so an excluded value disappears from the answer.
- Using only the endpoints to find a range when a turning point lies inside the interval.
- Reading fg(x) the wrong way round, applying f first instead of g.
- Treating f⁻¹(x) as 1/f(x), which is a different function altogether.
- Finding an inverse but forgetting its domain, or choosing a restriction that does not sit on one side of the turning point.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before opening the answer, and write every restriction as a separate line. A domain statement missing from your working is a lost mark even when the algebra is correct. Use the function composition and inverse explorer to test an input or a graph after you have tried it by hand, not before.
When you get something wrong, read the routing list at the end of the practice set and return to the lesson it names. Keep a record of the error types in the mistake log and retest queue, and retry a fresh question a few days later.
If your progress stalls on the same habit, a teacher can look at your written solutions in online one-to-one Additional Mathematics tuition.