A function gives exactly one output for each input. This lesson teaches you to read the notation, find a value and state the range without guessing.
It is the first lesson in SPM Additional Mathematics functions. The next lesson, composite functions in the right order, relies on it.
What does the notation say?
Take f(x) = 3x − 2 with domain {0, 1, 2, 3}. Each symbol has a job.
| Symbol | Name | In this example |
|---|---|---|
| x | object or input | 0, 1, 2 or 3 |
| f(x) | image or output | the value 3x − 2 gives |
| {0, 1, 2, 3} | domain | the inputs allowed |
| {−2, 1, 4, 7} | range | the outputs that actually occur |
The range comes from substituting each allowed input: f(0) = −2, f(1) = 1, f(2) = 4, f(3) = 7. The notation f: x → 3x − 2 says the same thing in words: “f sends x to 3x − 2”.
Worked example: endpoints are not always enough
Let g(x) = x² with domain −2 ≤ x ≤ 3. Find the range.
A common first attempt substitutes the endpoints: g(−2) = 4 and g(3) = 9, then writes 4 ≤ g(x) ≤ 9. That misses the turn in the graph.
Ask what x² does inside the interval. The interval includes x = 0, where g(0) = 0, and squaring never produces anything lower. The smallest output is 0. The largest is 9, from x = 3, because 9 is bigger than g(−2) = 4.
So the range is 0 ≤ g(x) ≤ 9. A quick sketch confirms it: the curve dips to the origin between the two endpoints.
The mistake that costs marks
The slip is to treat a restricted domain as if every function were a straight line, where the endpoints always give the smallest and largest outputs. For a straight line that is true. For x², |x| and other turning rules it is not.
The fix is a two-step habit. First, write the endpoint values. Second, ask whether the rule turns around between them, and if it does, add the turning value to your list before picking the lowest and highest.
Check yourself
Let h(x) = |x − 2| with domain 0 ≤ x ≤ 5. Find h(0), h(5) and the range.
Answer
h(0) = |−2| = 2 and h(5) = |3| = 3.
The rule turns at x = 2, where h(2) = 0. That value lies inside the domain, so the smallest output is 0. The largest is 3.
The range is 0 ≤ h(x) ≤ 3. Checking only the endpoints would have given 2 ≤ h(x) ≤ 3, which is wrong.
What to study next
Continue to composite functions, then test the chapter with the functions practice set. Record every slip in the mistake log and paper-error review.
If you want a teacher to watch how you read a question, see online one-to-one Additional Mathematics tuition.