IB Maths AA HL Topic 2 — Functions Paper 1 & 2 ~8 min read

Language of Functions

A function is a rule that takes an input and gives back exactly one output. The vocabulary around them — domain, range, mapping, function notation — is what every later topic builds on. Get fluent with the language now and the rest of the chapter is mostly mechanical.

📘 What you need to know

A function takes each input to exactly one output. So functions can be one-to-one or many-to-one — but never one-to-many.
Vertical line test: a sketch is a function if every vertical line crosses it at most once.
Notation: f(x) means “apply f to input x“. So f(3) means substitute x = 3.
Domain = set of allowed inputs (in terms of x). Range = set of possible outputs (in terms of f(x)).
Largest possible domain is usually all real numbers (ℝ) except values that break the function — divide-by-zero, square root of a negative, log of zero or negative.
Number sets: ℕ (naturals 0, 1, 2, …), ℤ (integers), ℚ (rationals), ℝ (reals). Each set is contained in the next: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.
Piecewise function: different rules apply on different intervals. May or may not be continuous at the boundaries.

Mappings — four types, only two are functions

A mapping links inputs to outputs. There are four flavours:

One-to-one ✓

f(x) = 2x + 5

function — each input has its own unique output

Many-to-one ✓

f(x) = x²

function — different inputs can share an output

One-to-many ✗

y² = x

not a function — one input gives multiple outputs

Many-to-many ✗

x² + y² = 25

not a function — circle fails the vertical line test

Vertical line test: draw any vertical line on the graph. If it ever crosses the curve more than once, it’s not a function. If every vertical line hits at most once, it is.

Function notation, domain & range

Function notation f(x) = “the output of f when the input is x”
f(3) ⟹ substitute x = 3 into the rule

Domain

set of inputs (in x)

e.g. x ≥ 0, or x ∈ ℝ, x ≠ 2

Range

set of outputs (in f(x))

e.g. f(x) ≥ 1, or f(x) ∈ ℝ

The point (a, b) is on the graph of y = f(x) precisely when f(a) = b. Inputs go on the x-axis, outputs on the y-axis.

If a domain isn’t written next to a function, assume it’s the largest possible — usually all real numbers, with whatever values break the rule excluded.

Number sets — symbols you’ll see in domains

ℕ — naturals

0, 1, 2, 3, …

whole non-negative numbers

ℤ — integers

…, −2, −1, 0, 1, 2, …

whole numbers + negatives

ℚ — rationals

a/b, b ≠ 0

fractions of integers

ℝ — reals

all of the above + π, √2, …

everything on the number line

You’ll also see ℤ⁺ for positive integers (1, 2, 3, …). The symbol “x ∈ ℝ” reads as “x is a real number”.

Largest possible domain — when do things break?

Start with all real numbers and exclude whatever breaks the rule:

Square root

√(expression)

need expression ≥ 0

Fraction (1 / something)

1expression

need expression ≠ 0

Logarithm

ln(expression)

need expression > 0

Polynomial / exponential

e.g. x³, e^x

no restrictions — x ∈ ℝ

Piecewise functions — different rules on different intervals

A piecewise function is a single function defined by different rules depending on which interval the input falls in. To evaluate f(a), check which interval a sits in, then apply the matching rule.

Example piecewise f(x) = { 3x − 2 if x ≤ 4 ; x² − 8 if x > 4 }

A piecewise function is continuous at a boundary if the two rules give the same value there. Otherwise the graph “jumps”.

The intervals must not overlap — every x belongs to exactly one piece. If they overlapped, you’d have ambiguity (and it wouldn’t be a function).

Worked examples

WE 1

Evaluate a function at a given input

Given f(x) = 3x² − 2x + 1, find f(−2).

Substitute x = −2 f(−2) = 3(−2)² − 2(−2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 f(−2) = 17 watch the sign on −2(−2) — two negatives multiply to give a positive

WE 2

Find the range of a linear function with a given domain

For f(x) = 2x − 5 with domain −3 ≤ x ≤ 4, find the range of f.

Linear, increasing → endpoints give endpoints f(−3) = 2(−3) − 5 = −11 f(4) = 2(4) − 5 = 3 range: −11 ≤ f(x) ≤ 3 for a strictly increasing linear function, the smallest input gives the smallest output

WE 3

Find the range when the vertex matters

For f(x) = x² − 4x + 7 with domain 1 ≤ x ≤ 5, find the range of f.

Step 1: Find vertex (minimum since a > 0) x = −b/(2a) = 4/2 = 2 (lies inside domain 1 ≤ x ≤ 5) f(2) = 4 − 8 + 7 = 3 → minimum value Step 2: Check endpoints for maximum f(1) = 1 − 4 + 7 = 4 f(5) = 25 − 20 + 7 = 12 → maximum range: 3 ≤ f(x) ≤ 12 for a quadratic, always check whether the vertex lies inside the domain — if yes, that’s an extremum

WE 4

Find the largest possible domain

State the largest possible domain for each function:
(a) f(x) = √(x − 3) (b) g(x) = 1x − 5 (c) h(x) = ln(x + 2)

(a) Square root → expression inside ≥ 0 x − 3 ≥ 0 → x ≥ 3 (a) x ≥ 3 (b) Fraction → denominator ≠ 0 x − 5 ≠ 0 → x ≠ 5 (b) x ∈ ℝ, x ≠ 5 (c) Logarithm → input strictly > 0 x + 2 > 0 → x > −2 (c) x > −2 log needs strict > 0 (not ≥) because ln(0) is undefined

WE 5

Evaluate a piecewise function and check continuity

The function f is defined by f(x) = { x + 5 if x < 0; x² + 5 if 0 ≤ x ≤ 3; 3x + 4 if x > 3 }.
(a) Find f(−2), f(2), f(4). (b) Determine whether f is continuous at x = 0 and at x = 3.

(a) Pick the right rule for each input f(−2): x < 0 → use x + 5 → f(−2) = 3 f(2): 0 ≤ x ≤ 3 → use x² + 5 → f(2) = 9 f(4): x > 3 → use 3x + 4 → f(4) = 16 f(−2) = 3, f(2) = 9, f(4) = 16 (b) Continuity = both pieces agree at the boundary at x = 0: x + 5 → 5 and x² + 5 → 5 ✓ continuous at x = 3: x² + 5 → 14 and 3x + 4 → 13 ✗ jump of 1 unit continuous at x = 0; not continuous at x = 3 the boundary value itself uses the rule that includes the equals sign — at x = 3, that’s x² + 5 → 14

WE 6

Classify mappings — function or not?

Classify each as one-to-one, many-to-one, one-to-many, or many-to-many. State whether each is a function.
(a) y = 4x − 7 (b) y = (x − 3)² (c) x = y² + 1

(a) Linear: each x → unique y, each y from one x (a) one-to-one — function ✓ (b) Parabola (∪-shape): each x → one y, but two x-values share the same y e.g. x = 1 and x = 5 both give y = 4 (b) many-to-one — function ✓ (c) Sideways parabola: solving for y gives y = ±√(x − 1) one input x gives two outputs (positive and negative roots) (c) one-to-many — NOT a function ✗ vertical line test: (a) and (b) pass, (c) fails — a vertical line at x = 5 cuts (c) at two points

💡 Top tips

Vertical line test is the fastest way to spot if a graph is a function.
Domain in x, range in f(x). Don’t mix them up — examiners do check the variable.
For a quadratic on a closed interval, always check whether the vertex is inside the domain — that gives the extreme value.
Three things break a function: divide by zero, square root of negative, log of zero or negative. Memorise these and you’ll handle most “largest possible domain” questions.
For piecewise, write out which rule you’re using before substituting. Picking the wrong piece is the most common slip.
Sketch the graph on your GDC when finding a range — it’s much easier to read off the highest and lowest values visually.
Continuity at a boundary: just check both rules at that point. Equal → continuous. Not equal → jump.

⚠ Common mistakes

Calling a one-to-many mapping a function. Each input must give exactly one output.
Forgetting the vertex when finding range of a quadratic on a closed interval. Endpoints aren’t always the extremes.
Writing range in x instead of f(x). Domain uses x; range uses f(x) (or whatever the function is called).
Including x = 0 in the domain of 1/x. Always exclude any value that makes a denominator zero.
For piecewise functions, picking the rule by guess instead of checking the interval. Always look at which interval contains the input first.
Using ≤ where the boundary is excluded. ln(x) needs x > 0, not x ≥ 0 — log of zero is undefined.
Treating “many-to-many” as a function with restrictions. It’s not a function in the first place — there’s no rule that gives a single output.

Now you’ve got the language locked down — function notation, domain, range, mappings, piecewise. Everything from here builds on these ideas. The next note tackles composite functions: what happens when you feed the output of one function as the input to another, like f(g(x)).

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