Ch 1 · 1 Functions · O Level Additional Mathematics 4037
Topic 1 of 6 in Algebra and Functions — O Level Additional Mathematics 4037 — 65 lessons.
Introduction
A function is a rule that gives exactly one output for each input, like a vending machine: press B4, always get the same snack. Additional Mathematics uses functions everywhere, so this lesson sets out the precise words and notation, how to undo a function (its inverse), how to chain two together (a composite), and what the modulus does to a graph. (Syllabus topic 1.)
Key ideas — Function, domain and range
A function maps each element of its domain (the allowed inputs) to exactly one output. The set of outputs is the range (or image set). A rule is a function if no input has two outputs; on a graph, every vertical line meets the curve at most once.
🧑🏫 Teacher's explanation
Think of a function like a vending machine. You press a specific button (the input), and you get exactly one specific snack (the output). If pressing the 'A1' button sometimes gives you chips and sometimes gives you a chocolate bar, the machine is broken—it is not a function. In math, we call the set of all possible buttons you can press the domain, and the set of all possible snacks you could receive the range.
📖 Definitions
🔍 Worked example — Easy
Let f(x) = 3x - 2 for x ∈ mathbbR. State the domain and range of f.
🔍 Worked example — Standard
Given g(x) = x^2 for x ∈ mathbbR, explain why g(x) is a many-one function and find the range.
🔍 Worked example — Exam-style
The function h(x) = 1/x-3 is defined for x ≠ 3. Find the range of h(x).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Notation
f(x) = 2e^x, or f : x mapsto lg x for x > 0. f^-1(x) is the inverse. fg(x) = f(g(x)): apply g first. f^2(x) = ff(x) = f(f(x)), not [f(x)]^2.
🧑🏫 Teacher's explanation
Think of a function as a mathematical machine. You drop an input (the x value) into the machine, and it spits out exactly one output. The set of all possible inputs you are allowed to use is called the domain, and the set of all possible outputs the machine can produce is the range.
📖 Definitions
🔍 Worked example — Standard
Find the inverse function f^-1(x) for f(x) = 2e^x.
🔍 Worked example — Exam-style
A function f(x) = x^2 has a domain of x ∈ mathbbR. Explain why f(x) does not have an inverse function.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Find f^-1(x). (b) State the domain of f^-1(x). [3 marks]
Key ideas — Inverse functions
Only a one–one function has an inverse. To explain why f has no inverse, say it is many–one and give two inputs with the same output (e.g. f(0) = f(4) = 0 for f(x) = x^2 - 4x). A domain is often restricted (e.g. to one side of a vertex) to make f one–one.
🧑🏫 Teacher's explanation
Think of a function like a machine that takes an input and produces a specific output. An inverse function is simply that machine running in reverse. If the original machine takes a '3' and turns it into a '10', the inverse machine takes that '10' and turns it back into the '3'.
📖 Definitions
🔍 Worked example — Easy
Given f(x) = 3x - 5 for x ∈ mathbbR, find f^-1(x) and state its domain.
🔍 Worked example — Standard
Explain why g(x) = x^2 has no inverse for x ∈ mathbbR.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Composite functions
gf exists only when every output of f is a valid input of g: the range of f must lie inside the domain of g. Then the domain of gf is (part of) the domain of f, and the range of gf is part of the range of g. Order matters: in general fg ≠ gf.
🧑🏫 Teacher's explanation
Think of a function like a machine in a factory. You put a raw material (the input x) into machine g, and it spits out a processed part (the output g(x)). A composite function fg(x) is like a production line: you take that processed part and immediately drop it into machine f. The output of the first machine becomes the input for the second.
📖 Definitions
🔍 Worked example — Easy
Given f(x) = x + 5 and g(x) = x^2, find fg(3).
🔍 Worked example — Standard
Let f(x) = 2x - 1 and g(x) = x^2. Find the expression for gf(x) and fg(x).
🔍 Worked example — Exam-style
Given f(x) = √(x) and g(x) = x - 4, find the domain of fg(x).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The graph of y = |f(x)|
|f(x)| is never negative. To sketch y = |f(x)|, draw y = f(x) and reflect every part below the x-axis in the x-axis; parts above stay put. Examples:
🧑🏫 Teacher's explanation
Think of the absolute value function |f(x)| as a 'positivity filter'. In mathematics, the absolute value of a number is its distance from zero, which can never be negative. When we apply this to a function, we are essentially saying: 'I only care about the magnitude, not the direction.'
📖 Definitions
🔍 Worked example — Easy
Sketch the graph of y = |2x - 5| for x ∈ mathbbR.
🔍 Worked example — Standard
Sketch y = |x^2 - 4|.
🔍 Worked example — Exam-style
Given f(x) = 3sin(x), sketch y = |f(x)| for 0 ≤ x ≤ 2π.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: composites and f²
f(x) = 3x - 1 and g(x) = x^2 + 2, for x ∈ mathbbR.
🔍 Worked example — Easy
Given f(x) = 2x + 5 and g(x) = x - 3, find fg(x) and f^2(x).
🔍 Worked example — Exam-style
Let f(x) = √(x+2) for x ≥ -2. Find f^2(x) and state its domain.
⚠️ Common errors
Worked examples — Example 2: range and inverse
f(x) = (x - 2)^2 + 1 for x ≥ 2. The least value is 1, so the range is f(x) ≥ 1, and f is one–one on this domain.
🔍 Worked example — Easy
Given f(x) = 3x - 5 for x ∈ mathbbR, find the range of f and the inverse function f^-1(x).
🔍 Worked example — Standard
A function is defined as f(x) = (x - 2)^2 + 1 for x ≥ 2. Find f^-1(x) and state its domain.
🔍 Worked example — Exam-style
Explain why g(x) = x^2 has no inverse for the domain x ∈ mathbbR.
⚠️ Common errors
Worked examples — Example 3: why there is no inverse
h(x) = x^2 - 4x for x ∈ mathbbR has no inverse because it is many–one: h(0) = h(4) = 0, so the inverse would have to send 0 to two different values.
🔍 Worked example — Easy
Explain why f(x) = 3x + 2 has an inverse.
🔍 Worked example — Standard
Show that g(x) = x^2 for x ∈ mathbbR has no inverse.
🔍 Worked example — Exam-style
The function h(x) = x^2 - 4x is defined for x ∈ mathbbR. Explain why h(x) does not have an inverse.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) gf(x) = 1/2x + 5; exclude x = -2.5. (2) range f(x) ≤ 4; f^-1(x) = -1 + √(4 - x) for x ≤ 4. (3) f^-1(x) = e^x - 3; range f^-1(x) > -3. (4) roots x = -2 and x = 4; y-intercept |-8| = 8; the part between -2 and 4 is reflected, with maximum point (1, 9). (5) e^2x - 1 = 5 ⇒ x = tfrac12(1 + ln 5) = 1.30.