Ch 2 · 1.2 Functions · A Level Mathematics 9709
Topic 2 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 76 lessons.
Introduction
A function is a machine: put a number in, get exactly one number out. A currency converter is a function; converting back again is its inverse; converting pounds to dollars and then dollars to rupees is a composition. This lesson gives the language Cambridge uses for functions and the four basic ways to move and stretch a graph. (Syllabus section 1.2, Paper 1.)
Key ideas — Function, domain and range
A function f assigns to each input x in its domain exactly one output f(x). The set of outputs is the range. Notation: f : x mapsto 2x + 3 or f(x) = 2x + 3, with the domain stated, e.g. "for x ∈ mathbbR" or "for x ≥ 3".
🧑🏫 Teacher's explanation
Think of a function like a vending machine. You put in a specific coin (the input, or x), and the machine gives you exactly one specific snack (the output, or f(x)). The set of all possible coins the machine accepts is the domain, and the set of all possible snacks it can actually dispense is the range.
📖 Definitions
🔍 Worked example — Easy
Given f(x) = 3x - 5 for x ∈ mathbbR, find the range of f.
🔍 Worked example — Standard
The function g(x) = x^2 - 4x + 7 is defined for the domain x ≥ 0. Find the range of g.
🔍 Worked example — Exam-style
Let f(x) = 2x+1/x-3 for x ≠ 3. Find the inverse function f^-1(x).
⚠️ Common errors
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🎯 Exam question
Key ideas — Composite functions
gf(x) means apply f first, then g: gf(x) = g(f(x)). With f(x) = 2x + 3 and g(x) = x^2:
🧑🏫 Teacher's explanation
Think of a function as a machine that takes an input and produces an output. A composite function is like an assembly line where the output of the first machine becomes the raw material for the second machine. If the first machine produces a 'size 10' shoe, but the second machine only accepts 'size 5' materials, the assembly line breaks down.
📖 Definitions
🔍 Worked example — Easy
Given f(x) = 3x - 1 and g(x) = x^2, find gf(x) and fg(x).
🔍 Worked example — Standard
Let f(x) = √(x) for x ≥ 0 and g(x) = x - 5 for all x ∈ mathbbR. Determine if gf(x) exists and find its expression.
🔍 Worked example — Exam-style
Function f is defined by f(x) = x^2 - 4 for x ∈ mathbbR. Function g is defined by g(x) = 2x + 1 for x ∈ mathbbR. Find the values of x for which gf(x) = 11.
⚠️ Common errors
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🎯 Exam question
Key ideas — One-one functions
A function is one-one if each output comes from exactly one input. On a graph: every horizontal line meets the curve at most once. f(x) = x^2 for x ∈ mathbbR is not one-one (f(2) = f(-2)), but f(x) = x^2 for x ≥ 0 is. Only one-one functions have an inverse, which is why questions restrict the domain (often to one side of a vertex).
🧑🏫 Teacher's explanation
Imagine a vending machine where every button corresponds to exactly one snack. If you press 'Button A' and get a bag of crisps, and then press 'Button A' again and get a chocolate bar, the machine is broken—it is not 'one-one'. In mathematics, a one-one function is a 'perfectly reliable' machine: every unique input x produces a unique output y. If two different inputs produced the same output, you wouldn't know which input to go back to if you tried to reverse the process.
📖 Definitions
🔍 Worked example — Easy
Determine if f(x) = 3x - 5 for x ∈ mathbbR is one-one, and find its inverse f^-1(x).
🔍 Worked example — Standard
The function g(x) = x^2 - 4 is defined for x ≥ 0. State whether g(x) is one-one and find g^-1(x).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Inverse functions
The inverse f^-1 undoes f: f^-1f(x) = x. To find it:
🧑🏫 Teacher's explanation
Think of a function as a machine that takes an input and produces a specific output. An inverse function is simply that machine running in reverse. If the original function takes you from point A to point B, the inverse function takes you from point B back to point A.
📖 Definitions
🔍 Worked example — Standard
Show that f(x) = x^3 + 2 is a one-to-one function and find f^-1(x).
🔍 Worked example — Exam-style
A function g(x) = 2x+1/x-3 is defined for x ≠ 3. Find g^-1(x) and state its domain.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Show that f is a one-to-one function. (b) Find f^-1(x) and state its domain. [4 marks]
Key ideas — Graphs of f and its inverse
The graph of y = f^-1(x) is the reflection of y = f(x) in the line y = x. A sketch should show the mirror line y = x and both curves, with matching points such as (a, b) on f and (b, a) on f^-1.
🧑🏫 Teacher's explanation
Think of a function as a machine that takes an input x and turns it into an output y. An inverse function, f^-1(x), is simply that machine running in reverse. If the original machine turns a 2 into a 10, the inverse machine must turn a 10 back into a 2.
📖 Definitions
🔍 Worked example — Easy
The function f(x) = 3x - 5 is defined for the domain x ∈ mathbbR. Find the expression for f^-1(x).
🔍 Worked example — Standard
A function g is defined by g(x) = x^2 + 3 for x ≥ 0. Show that g is one-to-one and find g^-1(x).
🔍 Worked example — Exam-style
The graph of f(x) passes through the point (2, 7). State the coordinates of the corresponding point on the graph of f^-1(x).
⚠️ Common errors
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🎯 Exam question
(a) Show that f is one-to-one. [2] (b) Find an expression for f^-1(x). [3] [5 marks]
Key ideas — Transformations of graphs
Use these exact words, with the vector or the factor and direction. For a combination such as y = 2f(x) - 3, work outwards from f(x): first the stretch (factor 2 parallel to the y-axis), then the translation beginpmatrix 0 \\ -3 endpmatrix. The order can matter: translating first and then stretching gives y = 2big(f(x) - 3big), a different graph.
🧑🏫 Teacher's explanation
Think of a function y = f(x) as a set of instructions that takes an input x and produces an output y. A transformation is like applying a 'filter' or a 'modifier' to those instructions. If you change the output (the y-value), you are affecting the graph vertically—moving it up/down or stretching it away from the x-axis. If you change the input (the x-value) before the function even sees it, you are affecting the graph horizontally—moving it left/right or stretching it away from the y-axis.
📖 Definitions
🔍 Worked example — Easy
The graph of y = f(x) is translated by the vector beginpmatrix 0 \ -3 endpmatrix. Write the equation of the new graph.
🔍 Worked example — Standard
The graph of y = x^2 is stretched by a factor of 3 parallel to the y-axis and then reflected in the x-axis. Write the equation of the resulting graph.
🔍 Worked example — Exam-style
The graph of y = f(x) is transformed into y = f(2x - 4). Describe the sequence of transformations applied to y = f(x).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: range and inverse
f(x) = x^2 - 6x + 11 for x ≥ 3. State the range of f and find f^-1(x).
🔍 Worked example — Easy
f(x) = 2x + 5 for x ≥ 0. State the range of f and find f^-1(x).
🔍 Worked example — Standard
f(x) = x^2 - 6x + 11 for x ≥ 3. State the range of f and find f^-1(x).
🔍 Worked example — Exam-style
The function g(x) = 3/x-4 is defined for x ≠ 4. Find g^-1(x) and state its domain.
⚠️ Common errors
Worked examples — Example 2: inverse of a rational function
h(x) = 2x + 3/x - 1 for x ≠ 1. Find h^-1(x).
🔍 Worked example — Easy
Find the inverse of f(x) = 3x - 5.
🔍 Worked example — Standard
Find h^-1(x) for h(x) = 2x + 3/x - 1, x ≠ 1.
🔍 Worked example — Exam-style
A function is defined as g(x) = 4/x+2. Find g^-1(x) and state its domain.
⚠️ Common errors
Worked examples — Example 3: describing a combined transformation
Describe a sequence of two transformations that maps y = x^2 onto y = 3(x+2)^2.
🔍 Worked example — Easy
Describe the single transformation that maps y = x^2 onto y = x^2 + 5.
🔍 Worked example — Standard
Describe the sequence of transformations that maps y = x^3 onto y = 2(x - 3)^3.
🔍 Worked example — Exam-style
Describe the sequence of transformations that maps y = √(x) onto y = √(2x + 4).
⚠️ Common errors
Common mistakes
Practice
Answers: (1) fg(x) = 3/x+2 - 1; gf(x) = 1/3x+1 (x ≠ -tfrac13). (2) k = -1; f^-1(x) = -1 + √(x + 4) for x ≥ -4. (3) -1 ≤ f(x) ≤ 5. (4) y(x+4) = x ⇒ x = 4y/1-y, so f^-1(x) = 4x/1-x for x < 1. (5) A stretch, factor tfrac12, parallel to the x-axis. (6) (a) (2, 1); (b) (5, 5); (c) (2, -5); (d) (5, 2).