IB Maths AI SL Topic 2 — Further Functions & Graphs Paper 1 & 2 Foundational ~8 min read

Functions & Mappings

A function is a mapping that takes each input to exactly one output. That “exactly one” rules out mappings like “the square root” (which gives +x AND −x) — not a function. This note covers the four mapping types, function notation, domain and range, and piecewise functions: the foundations for every chapter ahead.

📘 What you need to know

Mapping: a rule that links each input to one or more outputs. There are four types: one-to-one, many-to-one, one-to-many, and many-to-many.
Function: a mapping where each input gives exactly one output. So only one-to-one and many-to-one mappings are functions.
Vertical line test: a graph represents a function if and only if every vertical line crosses it at most once.
Notation: if x is the input then f(x) is the output. So f(2) = 5 means input 2 gives output 5, plotted as the point (2, 5).
Domain: the set of all valid inputs — expressed in terms of x. Range: the set of all outputs — expressed in terms of f(x).
Piecewise function: defined by different rules on different intervals of x. Pick the rule whose interval contains your input, then substitute.

Mappings vs. functions — the four types

A mapping falls into one of four types — only the top two are functions, since functions can’t give multiple outputs for the same input.

Functions are mappings where each input has exactly one output. Top row (teal) qualify as functions; bottom row (red) don’t because an input gives more than one output.

The vertical line test

Quick visual check: draw any vertical line on the graph. If it ever crosses the curve at two or more points, that x-value has multiple outputs — not a function. y = x² passes; the sideways parabola x = y² fails.

Function notation, domain & range

Function notation & sets f(x) = output when input is x
Domain = {valid inputs x} | Range = {outputs f(x)}

Inputs go on the x-axis, outputs on the y-axis: f(3) = 7 means (3, 7) is on the graph. To find the range, check the function’s values across the stated domain — including endpoints and any turning points inside.

Number-set symbols: ℝ (reals), ℚ (rationals), ℤ (integers), ℤ⁺ (positive integers), ℕ (naturals). “Largest possible domain” usually means x ∈ ℝ with any forbidden values excluded (e.g. zero in a denominator, negatives inside an even root).

Quick trick: to find a range from a closed domain, evaluate the function at both endpoints and at any vertex/turning point inside the domain. The smallest of those is the min, the largest is the max.

Piecewise functions

A piecewise function uses different rules on different intervals of x. To evaluate: find which interval contains your input, then substitute into that rule. Intervals don’t overlap, but the function may not be continuous where they meet — plug the boundary into both adjacent rules to check.

🧭 Recipe — working with any function question

Identify the rule: what does the function do to its input? (e.g. “square it, then add 3”). For piecewise, identify which rule’s interval contains the input first.
Check the domain: confirm your input is allowed. Watch for division by zero, square roots of negatives, and explicit stated restrictions like 2 ≤ x ≤ 10.
Substitute: replace every x in the rule with the input value and simplify. f(3) means “wherever I see x, write 3″.
For range questions: check the function value at both endpoints of the domain and at any vertex/turning point inside the domain. Compare to find min and max.
Express clearly: write the domain using x (e.g. x ≥ 0) and the range using f(x) (e.g. f(x) ≥ 1). Don’t mix the two up.

Worked examples

WE 1

Evaluate a function in context

A car rental company charges customers using the function f(d) = 30 + 0.25d, where d is the distance driven in km and f(d) is the cost in pounds. Find the cost of driving 180 km.

Step 1 — substitute d = 180 into the rule f(180) = 30 + 0.25 × 180 Step 2 — simplify = 30 + 45 = 75 f(180) = £75 notation: f(180) means “output when input is 180”, NOT “f times 180”. Substitute, never multiply.

WE 2

Classify mappings as functions or not

For each mapping below, state which type it is (one-to-one, many-to-one, one-to-many, many-to-many) and whether it is a function.
(a) x → x² + 1 (b) x → “the numbers whose square is x” (c) x → 3x − 2

(a) x → x² + 1 x = 2 → 5, x = −2 → 5 (two inputs → same output) type: MANY-TO-ONE, FUNCTION ✓ (b) x → numbers whose square is x x = 9 → +3 AND −3 (one input → two outputs) type: ONE-TO-MANY, NOT a function ✗ (c) x → 3x − 2 linear — every input gives a unique output type: ONE-TO-ONE, FUNCTION ✓ (a) function · (b) NOT · (c) function vertical line test: (a) parabola passes, (b) sideways parabola fails, (c) straight line passes.

WE 3

Range of a linear function on a closed interval

The function g is defined by g(x) = −2x + 7 with domain 1 ≤ x ≤ 5. Find the range of g.

Step 1 — linear with negative slope, so it’s strictly decreasing max is at the LEFT endpoint, min is at the RIGHT endpoint Step 2 — evaluate at both endpoints g(1) = −2(1) + 7 = 5 (max) g(5) = −2(5) + 7 = −3 (min) −3 ≤ g(x) ≤ 5 for a LINEAR function on a closed interval, the range is always between the two endpoint values — no turning points to check.

WE 4

Range of a quadratic on a closed interval

The function h is defined by h(x) = x² − 6x + 10 with domain 1 ≤ x ≤ 7. Find the range of h.

Step 1 — find the vertex (turning point) x-coord: x = −b/(2a) = 6/2 = 3 3 IS inside [1, 7] — so check it Step 2 — evaluate at vertex and both endpoints h(3) = 9 − 18 + 10 = 1 (min, since a > 0) h(1) = 1 − 6 + 10 = 5 h(7) = 49 − 42 + 10 = 17 (max) 1 ≤ h(x) ≤ 17 for a quadratic, ALWAYS check if the vertex is inside the domain. If yes, it’s the min (a > 0) or max (a < 0); the other extreme comes from an endpoint.

WE 5

Piecewise function — mobile data plan

A mobile data plan charges C(g) pounds for g GB of data used in a month, where
C(g) = 12 if 0 ≤ g ≤ 5,
C(g) = 12 + 3(g − 5) if 5 < g ≤ 15,
C(g) = 42 + 5(g − 15) if g > 15.
Find C(3), C(10) and C(20).

Pick the rule whose interval contains the input g = 3: in [0, 5] → first rule C(3) = 12 g = 10: in (5, 15] → second rule C(10) = 12 + 3(10 − 5) = 12 + 15 = 27 g = 20: in (15, ∞) → third rule C(20) = 42 + 5(20 − 15) = 42 + 25 = 67 C(3) = £12, C(10) = £27, C(20) = £67 pick the interval FIRST, then substitute. Boundary points (5 and 15) belong to exactly one rule — check the inequalities so you don’t double-count.

WE 6

Largest possible domain — restriction from formula

The function f is given by f(x) = 1x² − 9. State the largest possible domain of f.

Step 1 — spot the restriction division by zero is undefined need: x² − 9 ≠ 0 Step 2 — solve x² − 9 = 0 to find forbidden x x² = 9 x = +3 or x = −3 Step 3 — exclude these from the reals x ∈ ℝ, x ≠ ±3 “largest possible domain” = all reals minus the formula’s bad values. Two killers: zero in a denominator, negatives inside an even root.

💡 Top tips

Domain ↔ x, Range ↔ f(x): write domain in x (e.g. x ≤ 5), range in f(x) (e.g. f(x) ≥ 1). Mixing them up costs marks.
Use your GDC for ranges: the graphing screen gives min and max instantly. Don’t try quadratic ranges in your head — the vertex is too easy to slip on.
Vertex inside the domain? If yes, vertex gives one extreme and an endpoint gives the other. If no, both extremes come from endpoints.
Piecewise: read inequalities carefully: “≤” includes the boundary, “<” excludes it. Input must fall in exactly one interval.
“Largest possible domain” = ℝ minus the bad values: scan for fractions (denominator = 0?) and even roots (inside < 0?).

⚠ Common mistakes

Treating f(x) as multiplication: f(3) is NOT “f times 3″ — it’s the output of f when input is 3. Substitute.
Calling a one-to-many mapping a function: functions are only one-to-one or many-to-one. So y = ±√x is NOT a function.
Forgetting the vertex for a quadratic range: just using endpoints misses the actual min/max if the turning point lies inside the domain. Always check it.
Using a piecewise rule whose interval doesn’t contain the input: e.g. plugging x = 12 into the rule for x ≤ 5. Find the correct interval first.
Confusing ℝ with ℤ: reals (ℝ) include all decimals; integers (ℤ) are whole numbers only.

Up next: Inverse Functions. An inverse reverses what the original does — if f(2) = 5, then f⁻¹(5) = 2. Inverses exist only for one-to-one functions, so the work classifying mappings pays off straight away. The graph of f⁻¹ is the mirror image of f in the line y = x.

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