Year 12 Maths - Methods (Unit 3 & Unit 4) Functions, relations and graphs

Identifying and describing relations and functions

Q: What is the difference between a relation and a function?

A relation is any set of ordered pairs (x, y). A function is a relation in which each x-value gives exactly one y-value, so no input is paired with two different outputs. Every function is a relation, but not every relation is a function. For example the set {(1, 2), (1, 5)} is a relation but not a function because the input 1 is paired with both 2 and 5.

Q: What is the vertical-line test?

The vertical-line test decides from a graph whether a relation is a function. If every vertical line meets the graph at most once, each x-value gives only one y-value and the graph is a function. If some vertical line meets the graph more than once, the relation is not a function; for example a circle fails the test.

Q: What do the domain and range of a function mean?

The domain is the set of all allowed input values, the x-coordinates of the ordered pairs. The range is the set of all resulting output values, the y-coordinates. For y equals x squared with x from minus 1 to 2, the domain is minus 1 to 2 and the range is 0 to 4.

Q: What does the notation f(x) mean?

f(x) is the output of the function f for the input x; it does not mean f multiplied by x. Writing y equals f(x) names the rule, and substituting a number evaluates it. If f(x) equals x squared minus 3x then f(2) equals 4 minus 6 equals minus 2.

Q: What is the difference between a one-to-one and a many-to-one function?

In a one-to-one function each output comes from exactly one input, so different inputs never share an output; this can be checked with the horizontal-line test. In a many-to-one function at least one output comes from several inputs, for example y equals x squared where x equals 2 and x equals minus 2 both give 4.

Q: How can you describe a relation?

A relation can be described in three main ways: by a rule such as y equals 2x plus 1, by listing the set of ordered pairs such as {(0, 1), (1, 3)}, or by drawing its graph. A mapping diagram, which draws arrows from each input to its output, is also useful for a small finite relation.

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Theory

A relation is any set of ordered pairs \((x,y)\); a function is a relation in which each \(x\)-value gives exactly one \(y\)-value. The vertical-line test checks this on a graph, while the domain and range record the allowed inputs and the resulting outputs — and a relation can be described by a rule, a set of pairs, or a graph.

A relation is a set of ordered pairs \((x,y)\). The domain is the set of all first coordinates (the inputs, or \(x\)-values) and the range is the set of all second coordinates (the outputs, or \(y\)-values). A relation may be given by a rule such as \(y=2x+1\), by listing the pairs such as \(\{(0,1),(1,3),(2,5)\}\), or by a graph.

A function is a special relation in which every input is paired with exactly one output — no \(x\)-value may give two different \(y\)-values. On a graph this is the vertical-line test: if every vertical line meets the graph at most once, the relation is a function. Functions are written with function notation \(y=f(x)\), where \(f(a)\) is the output when \(x=a\); for example if \(f(x)=x^{2}\) then \(f(3)=9\).

A function is one-to-one if each output comes from exactly one input (it also passes the horizontal-line test), and many-to-one if some output comes from several inputs — for instance \(y=x^{2}\) is many-to-one because \(x=2\) and \(x=-2\) both give \(4\). A mapping diagram, drawing an arrow from each input to its output, makes this structure clear for a small finite relation.

Key idea. A relation is a function precisely when every \(x\) maps to exactly one \(y\). Use the vertical-line test to decide "function or not", and the horizontal-line test to decide "one-to-one or many-to-one".

Passes the test: every vertical line meets \(y=x^{2}\) once, so it is a function

Fails the test: the line \(x=2\) meets \(x=y^{2}\) twice, so it is a relation but not a function

The domain and range of a relation, written in set-builder form:

\[\text{dom}=\{\,x : (x,y)\text{ is in the relation}\,\}, \qquad \text{ran}=\{\,y : (x,y)\text{ is in the relation}\,\}\]

dom = {x : (x, y)}, ran = {y : (x, y)}

Function notation names the rule and lets you evaluate an output:

\[y=f(x),\qquad f:A\to B,\ x\mapsto f(x) \qquad\text{e.g. } f(x)=2x+1\Rightarrow f(3)=7\]

y = f (x), f (3) = 2 (3) + 1 = 7

The single-valued condition (function) and the one-to-one condition:

\[\big(x,y_{1}\big),\big(x,y_{2}\big)\in f\ \Rightarrow\ y_{1}=y_{2} \qquad\qquad f(a)=f(b)\ \Rightarrow\ a=b\ \ (\text{one-to-one})\]

f (a) = f (b) \Rightarrow a = b

Reading a graph. Vertical line meets the graph more than once \(\Rightarrow\) not a function; horizontal line meets it more than once \(\Rightarrow\) many-to-one. The domain is the spread of the graph along the \(x\)-axis, the range its spread along the \(y\)-axis.

How to identify and describe a relation or function

Read the relation. Note whether it is given as a rule, a listed set of pairs, or a graph.
Function or not? Check that each \(x\)-value gives only one \(y\)-value — on a graph apply the vertical-line test; from a rule, see whether solving for \(y\) gives a single value (e.g. \(y=\pm\sqrt{\;}\) means two).
Classify. If it is a function, decide one-to-one or many-to-one using the horizontal-line test.
State domain and range. The domain is every allowed \(x\); the range is every resulting \(y\). Watch for restrictions such as division by zero or a negative under a square root.
Describe it. Give the rule, the set of pairs, or the graph as required, using function notation \(y=f(x)\) when it is a function.

Quick shortcut. A single unbroken curve that never doubles back over an \(x\)-value is a function; if the same \(x\) ever produces two \(y\)-values (like a circle or a sideways parabola), it is only a relation.

Example 1 — a listed set of pairs

Decide whether \(\{(-2,4),(0,0),(1,1),(3,9)\}\) is a function, and state its domain and range.

Solution

Check the inputs — no \(x\)-value is repeated:

\(x\)-values

\(=\)

\(-2,\ 0,\ 1,\ 3\) (all different)

Each input has exactly one output, so it is a function. Read off domain and range:

domain	\(=\)	\(\{-2,0,1,3\}\)
range	\(=\)	\(\{0,1,4,9\}\)

\(\therefore\) it is a function; domain \(\{-2,0,1,3\}\), range \(\{0,1,4,9\}\)

dom = {- 2, 0, 1, 3}

Example 2 — function notation

For \(f(x)=x^{2}-3x\), find \(f(0)\), \(f(2)\) and \(f(-1)\), then solve \(f(x)=0\).

Solution

Substitute each value into the rule:

\(f(0)\)	\(=\)	\(0^{2}-3(0)=0\)
\(f(2)\)	\(=\)	\(4-6=-2\)
\(f(-1)\)	\(=\)	\(1+3=4\)

Solve \(f(x)=0\) — factorise:

\(x^{2}-3x\)	\(=\)	\(x(x-3)=0\)
\(x\)	\(=\)	\(0\ \text{ or }\ 3\)

\(\therefore\) \(f(0)=0,\ f(2)=-2,\ f(-1)=4\); and \(f(x)=0\) at \(x=0,3\)

f (2) = 4 - 6 = - 2

Example 3 — reading a graph

The graph shows \(y=x^{2}\) for \(-1\le x\le 2\). Explain why it is a function, and state its domain and range.

Solution

Function test — every vertical line meets the curve once:

vertical-line test

\(\Rightarrow\)

one \(y\) per \(x\), so a function

Domain is the \(x\)-spread; range runs from the vertex value \(0\) up to the highest point:

domain	\(=\)	\([-1,\,2]\)
range	\(=\)	\([0,\,4]\)

\(\therefore\) a (many-to-one) function; domain \([-1,2]\), range \([0,4]\)

ran = [0, 4]

Example 4 — a rule that is not a function

Does \(x^{2}+y^{2}=9\) define \(y\) as a function of \(x\)? State the domain and range of the relation.

Solution

Solve for \(y\) to see how many outputs each \(x\) gives:

\(y^{2}\)	\(=\)	\(9-x^{2}\)
\(y\)	\(=\)	\(\pm\sqrt{9-x^{2}}\)

The \(\pm\) gives two \(y\)-values (e.g. \(x=0\Rightarrow y=\pm 3\)), so it fails the vertical-line test:

\(x=0\)

\(\Rightarrow\)

\(y=3\text{ or }y=-3\) — not a function

Domain needs \(9-x^{2}\ge 0\); the circle has radius \(3\), so:

domain	\(=\)	\([-3,\,3]\)
range	\(=\)	\([-3,\,3]\)

\(\therefore\) a relation but not a function; domain \([-3,3]\), range \([-3,3]\)

y = \pm \sqrt{9 - x^{2}}

Common pitfalls

Repeated \(y\)-values are fine; repeated \(x\)-values are not. \(\{(1,2),(3,2)\}\) is a function (two inputs may share an output), but \(\{(1,2),(1,5)\}\) is not, because the input \(1\) has two outputs.

Vertical for "function", horizontal for "one-to-one". The vertical-line test decides whether a graph is a function; the horizontal-line test decides whether a function is one-to-one. Do not swap them.

\(f(x)\) is not multiplication. \(f(x)\) means the output of \(f\) at \(x\), not \(f\times x\). Also remember to exclude \(x\)-values that break the rule, such as division by zero or a negative under a square root.

Frequently asked questions

What is the difference between a relation and a function?

A relation is any set of ordered pairs \((x,y)\); a function is a relation in which each \(x\)-value gives exactly one \(y\)-value. Every function is a relation, but \(\{(1,2),(1,5)\}\) is a relation that is not a function.

What is the vertical-line test?

If every vertical line meets a graph at most once, each \(x\) gives only one \(y\) and the graph is a function. If some vertical line cuts it more than once — as a circle does — it is not a function.

What do the domain and range of a function mean?

The domain is the set of allowed inputs (\(x\)-values); the range is the set of resulting outputs (\(y\)-values). For \(y=x^{2}\) with \(-1\le x\le 2\), the domain is \([-1,2]\) and the range is \([0,4]\).

What does the notation f(x) mean?

\(f(x)\) is the output of \(f\) for the input \(x\) — not \(f\times x\). If \(f(x)=x^{2}-3x\) then \(f(2)=4-6=-2\).

What is the difference between a one-to-one and a many-to-one function?

One-to-one: each output comes from exactly one input (it passes the horizontal-line test). Many-to-one: some output comes from several inputs, such as \(y=x^{2}\) where \(x=2\) and \(x=-2\) both give \(4\).

How can you describe a relation?

By a rule such as \(y=2x+1\), by listing its ordered pairs such as \(\{(0,1),(1,3)\}\), or by drawing its graph. A mapping diagram of arrows from inputs to outputs also works for a small finite relation.

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Implied domains and types of functions

Identifying and describing relations and functions

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How to identify and describe a relation or function

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