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Cambridge IGCSE · Topic 1.2
IGCSE Add Maths 0606 / Topic 1 / IGCSE

One input, one output.

A function is a rule that sends every input in its domain to exactly one output. Learn the words examiners use (domain, range, one-one, many-one), read and write notation such as \(f(x) = 2e^{x}\) and \(f : x \mapsto \lg x\), and find the range of a function, including a quadratic on a restricted domain.

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The key idea

A function maps every input in its domain to exactly ONE output. The range is the set of outputs, written in terms of \(f(x)\), e.g. \(f(x) \ge -3\). A function is one-one if each output comes from only one input, and many-one if some output comes from two or more inputs.

x y −3 −1 2 3 −2 −6 (−3, −2) (2, 3) minimum (−1, −6) range −6 ≤ f(x) ≤ 3
Fig. 1: f(x) = (x + 1)² - 6 for -3 ≤ x ≤ 2. The least output is at the vertex inside the domain, not at an end, so the range is -6 ≤ f(x) ≤ 3
Section 01

What makes a function?

A function is a rule that takes each input from a set called the domain and gives exactly ONE output. In the lab, pick a rule and watch the mapping diagram: every arrow starts at an input and ends at a single output. Two inputs may share an output, but one input may never split into two outputs.

MappingExampleA function?
one-one\(x \mapsto 2x + 1\): each output from one inputyes, a one-one function
many-one\(x \mapsto x^{2}\): \(-3\) and \(3\) both give \(9\)yes, a many-one function
one-many\(x \mapsto \pm\sqrt{x}\): \(4\) gives \(2\) and \(-2\)no
Worked example Function or not, and which type?
For each rule on the given domain, say whether it is a function and, if so, whether it is one-one or many-one. (a) \(x \mapsto 5 - 3x\), \(x \in \mathbb{R}\) (b) \(x \mapsto x^{2} + 1\), \(x \in \mathbb{R}\) (c) \(x \mapsto \pm\sqrt{x}\), \(x \ge 0\)
  1. (a) Each \(x\) gives one value of \(5 - 3x\), and the graph is a straight line that always decreases, so different inputs give different outputs: a one-one function.
  2. (b) Each \(x\) gives one value of \(x^{2} + 1\), but \(x = -2\) and \(x = 2\) both give \(5\): a many-one function.
  3. (c) The input \(4\) gives two outputs, \(2\) and \(-2\), so this rule is not a function.
Answer: (a) one-one function   (b) many-one function   (c) not a function
Worked example Explain it in words
The function f is defined by \(f : x \mapsto x^{2} - 4x\) for \(x \in \mathbb{R}\). Explain why f is a function, and explain why it is not one-one.
  1. For every real number \(x\), the expression \(x^{2} - 4x\) has exactly one value, so each input has exactly one output: f is a function.
  2. \(f(0) = 0\) and \(f(4) = 16 - 16 = 0\).
  3. Two different inputs give the same output, so f is many-one, not one-one.
Answer: Each \(x\) gives one output, so f is a function; \(f(0) = f(4) = 0\), so it is many-one
Examiner trap

'Function' and 'one-one' are different tests. A function needs each INPUT to have one output; one-one needs each OUTPUT to come from one input. \(x \mapsto x^{2}\) is still a function even though it is many-one.

Section 02

Reading function notation.

The same function can be written \(f(x) = 2e^{x}\) or \(f : x \mapsto 2e^{x}\), read 'f maps \(x\) to \(2e^{x}\)'. The domain is stated alongside, e.g. 'for \(x \in \mathbb{R}\)' (all real numbers) or 'for \(x > 0\)'. Here \(\lg x\) means \(\log_{10} x\) and \(\ln x\) means \(\log_{e} x\). To find \(f(a)\), replace every \(x\) with \(a\).

Worked example Exponential and logarithm
\(f(x) = 2e^{x}\) for \(x \in \mathbb{R}\) and \(g : x \mapsto \lg x\) for \(x > 0\). Find (a) \(f(0)\) (b) \(f(\ln 3)\) (c) \(g(100)\) (d) the value of \(x\) for which \(g(x) = -1\).
  1. (a) \(f(0) = 2e^{0} = 2 \times 1 = 2\).
  2. (b) \(e^{\ln 3} = 3\), so \(f(\ln 3) = 2 \times 3 = 6\).
  3. (c) \(g(100) = \lg 100 = \lg 10^{2} = 2\).
  4. (d) \(\lg x = -1\) means \(x = 10^{-1}\).
Answer: (a) \(2\)   (b) \(6\)   (c) \(2\)   (d) \(x = \frac{1}{10}\)
Worked example Substituting an expression
\(f(x) = x^{2} - 3x\) for \(x \in \mathbb{R}\). Find \(f(a + 1)\) in terms of \(a\), and solve \(f(a + 1) = f(a)\).
  1. \(f(a + 1) = (a + 1)^{2} - 3(a + 1) = a^{2} + 2a + 1 - 3a - 3 = a^{2} - a - 2\).
  2. \(f(a) = a^{2} - 3a\), so \(a^{2} - a - 2 = a^{2} - 3a\).
  3. \(2a = 2\), so \(a = 1\). Check: \(f(2) = -2\) and \(f(1) = -2\).
Answer: \(f(a + 1) = a^{2} - a - 2\); \(a = 1\)
Your turn
\(h(x) = 3e^{x} - 1\) for \(x \in \mathbb{R}\). Find \(h(\ln 2)\) and solve \(h(x) = 14\), giving your answer in terms of a logarithm.
Examiner trap

\(f(2x)\) is not \(2f(x)\). With \(f(x) = 2e^{x}\), \(f(2x) = 2e^{2x}\) but \(2f(x) = 4e^{x}\). Replace the \(x\) inside the rule; do not multiply the answer.

Stage 1 · Learn

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Four quick checks. Each correct answer earns XP and records this skill on your profile.

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Which of these is a many-one function?

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\(f(x) = 2e^{x}\). Find \(f(\ln 5)\).

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What is the largest possible domain of \(f(x) = \sqrt{x - 3}\)?

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Find the range of \(f(x) = x^{2} - 2\) for \(x \in \mathbb{R}\).

Section 03

Domain and range.

The domain is given in the question, but sometimes you must find the largest domain a rule allows. The range is every output the domain produces. For a quadratic, complete the square to find the vertex, then check whether the vertex lies inside the domain as well as working out the values at the ends (Fig. 1).

Worked example A quadratic on a restricted domain
Find the range of \(f(x) = x^{2} + 2x - 5\) for \(-3 \le x \le 2\).
  1. Complete the square: \(x^{2} + 2x - 5 = (x + 1)^{2} - 1 - 5 = (x + 1)^{2} - 6\).
  2. The vertex is \((-1, -6)\), and \(x = -1\) lies inside the domain, so the least value is \(-6\).
  3. Ends: \(f(-3) = 4 - 6 = -2\) and \(f(2) = 9 - 6 = 3\). The greater is \(3\).
Answer: \(-6 \le f(x) \le 3\)
Worked example Exponential and logarithm ranges
Find the range of (a) \(f(x) = 2e^{x} + 1\) for \(x \in \mathbb{R}\) (b) \(g : x \mapsto \lg x\) for \(x \ge 10\).
  1. (a) \(e^{x} > 0\), so \(2e^{x} > 0\) and \(2e^{x} + 1 > 1\). As \(x\) increases, \(f(x)\) grows without limit.
  2. The value 1 is never reached, so the inequality is strict.
  3. (b) \(\lg x\) increases with \(x\), and \(g(10) = \lg 10 = 1\), so the outputs start at 1 and keep increasing.
Answer: (a) \(f(x) > 1\)   (b) \(g(x) \ge 1\)
Your turn
Find the range of \(f(x) = 3 - 2x\) for \(-1 < x \le 4\).
Examiner trap

Do not just substitute the ends. For Fig. 1, the ends alone give \(-2 \le f(x) \le 3\) and miss every output from \(-6\) to \(-2\). Always look for a vertex inside the domain, and write the answer with \(f(x)\), not \(x\).

Stage 2 · Exam

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Find the range of \(f(x) = x^{2} - 8x + 10\) for \(1 \le x \le 6\).

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The function f is defined by \(f : x \mapsto \lg(2x + 4)\). State the largest possible domain of f.

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The function f is defined by \(f(x) = 3e^{x} - 2\) for \(x \in \mathbb{R}\). State the range of f.

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The function f is defined by \(f(x) = (x - 2)^{2} + 1\) for \(0 \le x \le 5\). Which statement is true?

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Six original Cambridge-style questions
Range of a linear function, exponential and logarithm notation, a range by completing the square, explaining in words why a rule is a function, and finding a constant from the least value of a quadratic.