Summary — Everything In This Chapter
- A function turns an input
x into exactly one output f(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.
Seeing the machine in action
Put in 3 → 2 × 3 + 1 = 7
Put in −4 → 2 × (−4) + 1 = −7
Put in the letter x (an unknown number) → the output is 2x + 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.
Don't mix these up!
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.
Substituting different things in
If the input is 2, the output is f(2)
If the input is m, the output is f(m)
If the input is t + 5, the output is f(t + 5) — you replace every x in the formula with the whole expression (t + 5)
2. Domain & Range