Functions
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Functions
Summary — Everything In This Chapter
- A function turns an input
xinto exactly one outputf(x)using fixed operations. - Function notation: f(x) = ... lets you substitute numbers or expressions in for x.
- The domain is the set of allowed inputs; the range is the set of resulting outputs.
- Certain inputs must be excluded from a domain — usually values that cause division by zero or square-rooting a negative number.
- A composite function like gf(x) means "do f first, then g" — read right to left, the letter closest to x goes first.
- An inverse function f⁻¹(x) undoes what f(x) does, using opposite operations in reverse order.
- The domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f.
1. Introduction to Functions
What is a function?
Think of a function as a little machine on a conveyor belt. You drop a number in one end (that's the input), the machine does the exact same set of mathematical operations to it every single time, and a number pops out the other end (that's the output).
For example, imagine a machine labelled "double the number and add 1." Two operations happen inside: multiply by 2, then add 1.
That last line is the important one: a function isn't really about one specific number, it's a rule that works for any input you throw at it — including letters and even whole expressions, as you'll see later.
Function notation
Instead of writing y = 2x + 1, we can give the rule a name — usually the letter f — and write f(x) = 2x + 1. Here, x is the input and f(x) is the output. You'll also see other letters used, like g, h, or j — typically a new letter signals a brand new rule/function.
f(x) = 15 means "an input of x produces an output of 15" — this is a statement you'd solve as an equation to find x.
f(15) means "substitute the number 15 into the function" — you calculate a number, you don't solve anything.
Students lose easy marks by treating these as the same instruction. They are opposites: one gives you a number to plug in, the other gives you a number to solve for.
How a function works with different inputs
A function has an input x and an output f(x). The magic trick is: whatever you put in place of x on the left, you put in place of every x on the right.
A function is defined as f(x) = 3x² − 2x + 1.
2. Domain & Range
Functions and graphs
Every function can be drawn as a graph. Just replace f(x) = ... with y = ..., and the x-axis becomes your set of inputs, while the y-axis becomes your set of outputs. This connection is exactly why sketching helps you "see" a function's behaviour instead of just guessing.
What is the domain?
The domain is the set of all inputs a function is allowed to take. Not every function accepts every number — some numbers would break the machine (like trying to divide by zero), and some domains are simply restricted on purpose because a question only cares about a certain range of inputs.
Domains always describe values of x — never y, never f(x). You can use "≠" (not equal to) or inequality signs to describe them.
Values that must be excluded
Two classic troublemakers show up again and again in IGCSE functions:
Compare this to something like f(x) = x² or f(x) = 3x + 2 — these two never break, no matter what you put in, so their domain is simply "all values of x."
What is the range?
The range is the set of all outputs a function actually produces, once you know the domain. Crucially — the range depends on the domain. Change what you're allowed to put in, and you change what comes out.
Ranges always describe values of f(x) — never x, never y. Same tools apply: "≠" and inequality signs.
Take f(x) = 3x + 2 with domain x > 0.
Take f(x) = x² with domain "all values of x."
Sketching a function is the single best way to "see" both its domain and range at once. If you know what the graph of y = 1/x, y = x², y = x³, or the trig graphs look like, you can read off the range almost instantly — including for transformed and restricted versions of them.
Two functions are given by f(x) = 10 − x and g(x) = 1 / (2x − 1).
When a domain flips a strict inequality (<) into a "not-strict" inequality (≤) for the range, it happens because of which end of the graph is increasing vs decreasing. Always check with a quick sketch or by testing both endpoints — never assume the inequality signs "carry over" unchanged.
3. Composite Functions
What is a composite function?
A composite function is what happens when you feed the output of one function straight into another function as its input — like two machines bolted together on the same conveyor belt. The number goes through machine 1, whatever pops out gets fed straight into machine 2, and only then do you get your final answer.
The notation — and why it's backwards from what you'd expect
Rewrite gf(x) as g(f(x)) in your head or on paper — literally add the brackets back in. Now it visually reads "g of f(x)," which makes it obvious f happens first (it's the inner bracket) and g happens second (it's applied to whatever f produced).
Also worth knowing: fg(x) means do g first, then f. And ff(x) — sometimes written f²(x) — means apply f to itself twice in a row. Be careful: f²(x) does not mean [f(x)]², i.e. it does not mean "square the output of f." It means "run the input through f, then run that result through f again."
Substituting numbers into composite functions
The strategy: put the number into whichever function is closest to it, get an output, then feed that output into the other function.
f(x) = 2x + 1 and g(x) = 1/x.
Notice gf(2) ≠ fg(2) — the order genuinely matters, exactly like putting socks on before shoes gives a very different result than shoes before socks!
Composite functions algebraically
Same idea, but now you substitute the entire expression for the inner function in place of every x in the outer function.
f(x) = 2x − 1 and g(x) = (x + 2)².
4. Inverse Functions
What is an inverse function?
An inverse function undoes exactly what the original function did — like hitting rewind on the machine. If a function "doubles the number, then adds 1," its inverse must "subtract 1, then halve the result." Notice the operations are reversed and the order is flipped — this is exactly the same logic you already use when solving equations or rearranging a formula.
If f(a) = b, then f⁻¹(b) = a. In other words, whatever came out of f, if you put that back into f⁻¹, you get back exactly what you started with. For example, if f(3) = 7, then f⁻¹(7) = 3.
Finding an inverse function algebraically
There's a clean, reliable 4-step method — memorise this exact sequence:
Step 2 — Swap every x and every y (letters change, nothing moves position)
Step 3 — Rearrange the new equation to make y the subject again
Step 4 — Replace y with f⁻¹(x) — your final answer should never contain the letter y
Find the inverse of f(x) = 2x + 1.
Inverse and composite functions together
Here's a beautiful shortcut: applying a function and then its inverse (in either order) always cancels out and just gives you back x, untouched.
This is incredibly useful when finding f⁻¹(x) algebraically would be too hard or even impossible (for example, with exponential functions before you've learned logarithms).
Solve f⁻¹(x) = 5, where f(x) = 2ˣ.
Domain and range of an inverse function
This is one of the most elegant (and most-tested) ideas in the whole chapter: the inverse function literally swaps the roles of domain and range.
Range of f⁻¹ = Domain of f
This makes total sense once you think about it: f⁻¹ takes f's outputs and turns them back into f's inputs. So whatever used to come out of f becomes what goes into f⁻¹, and vice versa.
Always write the domain of f⁻¹ in terms of x (e.g. "x > 5"), and always write the range of f⁻¹ in terms of f⁻¹(x) (e.g. "f⁻¹(x) > 5") — even though the numbers are identical to f's range and domain, the letters used to describe them must change to match whichever function you're now describing.
A function f(x) = 5 − 3x has the domain −2 < x ≤ 7.
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
The single most common error: students calculate fg(x) as if it means "f, then g" because that's the order the letters appear. It's the opposite! gf(x) means f happens first. Always rewrite gf(x) as g(f(x)) before you start — the brackets don't lie.
When a denominator is a product like (x − 1)(x + 7), students often only spot one root and forget the other. Set the ENTIRE denominator equal to zero and solve for all values — every root must be excluded from the domain.
When converting a domain into a range, don't assume the inequality signs stay in the same place. Always test both endpoints of the domain, work out which output is bigger, and pay attention to whether each endpoint is "equal to" (≤/≥) or "strictly less/greater than" (</>) in the original domain — carry that distinction through correctly.
The "−1" in f⁻¹(x) is not an exponent — it's just notation meaning "the inverse function." Never confuse it with 1 divided by f(x). They are completely different objects.
When finding an inverse, it's easy to just rearrange for x without ever swapping x and y first. Remember: Step 2 (swap) MUST happen before Step 3 (rearrange), or your final answer will be wrong even though your algebra is technically correct.
Expect combined questions: "Find gf(x), then state its domain and range" — testing composite functions AND domain/range together in one question. Also very common: "Use ff⁻¹(x) = x to solve f⁻¹(x) = [number]" for functions where finding f⁻¹ directly is hard (exponentials, etc.) — this shows up specifically to test whether you understand the cancelling relationship, not just mechanical algebra.
- 2. Domain & Range
- Exam Tips & Common Mistakes
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