Functions
Understand functions for Queensland Year 11 Mathematical Methods (QCAA). A function is a special relation in which every input gives exactly one output — a rule with no ambiguity about the result.
You will learn to use the vertical line test to distinguish a function from a relation, check sets of ordered pairs, and classify a function as one-to-one or many-to-one — core ideas underpinning almost every topic that follows.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1) a function is a special relation in which each \(x\)-value gives exactly one \(y\)-value. This page shows how to use the vertical-line test to decide whether a graph is a function, how to test a set of ordered pairs, and how to classify a function as one-to-one or many-to-one and state its natural domain.
A relation is any set of ordered pairs. It is a function when every input \(x\) is paired with exactly one output \(y\) — no \(x\) may map to two different \(y\)-values.
The vertical-line test makes this visual: if every vertical line cuts the graph at most once, the graph is a function; if some vertical line cuts it twice or more, it is only a relation. A circle fails because a vertical line through it meets two points.
A function is one-to-one when different inputs always give different outputs (every horizontal line cuts once), and many-to-one when two inputs can share an output (some horizontal line cuts more than once). The natural domain is the largest set of \(x\) for which the rule is defined.
A relation \(f\) is a function exactly when:
The two graphical tests:
How to decide whether a relation is a function
- Graph: imagine sliding a vertical line across; if it ever meets the graph more than once, it is not a function.
- Ordered pairs: check for a repeated \(x\)-value with two different \(y\)-values; if none, it is a function.
- Classify: for a function, apply the horizontal-line test — one hit everywhere means one-to-one, otherwise many-to-one — then state the natural domain.
Look for a repeated input with different outputs:
| \(x=0\) | \(\to\) | \(y=-1\) |
| \(x=0\) | \(\to\) | \(y=1\) |
The input \(0\) is paired with both \(-1\) and \(1\), two different outputs.
Not a function — the input \(0\) maps to two \(y\)-values.
Pick any vertical line, say \(x=1.2\), and count the crossings:
| \(x=1.2\) | \(\Rightarrow\) | \(y=(1.2)^2\) |
| \(=\) | \(1.44\) |
Every vertical line meets the parabola at exactly one point.
\(y=x^2\) is a function (each \(x\) gives one \(y\)).
Solve for \(y\) and test a vertical line, say \(x=1\):
| \(y^2\) | \(=\) | \(9-x^2\) |
| \(y\) | \(=\) | \(\pm\sqrt{9-1}\) |
| \(=\) | \(\pm\sqrt{8}\) |
The line \(x=1\) meets the circle at two points, \((1,\sqrt{8})\) and \((1,-\sqrt{8})\).
No — the circle is a relation but not a function.
Apply the horizontal-line test — solve \(f(x)=-0.6\) as a sample level:
| \(x^2-4x+3\) | \(=\) | \(-0.6\) |
| \(x^2-4x+3.6\) | \(=\) | \(0\) |
| \(x\) | \(\approx\) | \(1.37\ \text{or}\ 2.63\) |
One horizontal line meets the parabola twice, so two inputs share an output.
Natural domain — a polynomial is defined for every real \(x\):
| \(\text{dom}\) | \(=\) | \(\mathbb{R}\) |
Many-to-one; natural domain \(\mathbb{R}\).
Common pitfalls
Frequently asked questions
What is the difference between a relation and a function?
A relation is any set of ordered pairs. A function is a relation in which every input \(x\) has exactly one output \(y\).
How does the vertical-line test work?
Slide a vertical line across the graph. If it ever meets the graph at two or more points, the graph is not a function; if it meets at most once everywhere, it is a function.
Why is a circle not a function?
A vertical line through a circle meets it at two points, giving one \(x\)-value two \(y\)-values, so \(x^2+y^2=r^2\) is a relation but not a function.
What does one-to-one and many-to-one mean?
One-to-one means every output comes from just one input (every horizontal line cuts once). Many-to-one means two inputs can share an output, like \(y=x^2\).
Can two ordered pairs have the same y-value?
Yes. Different inputs may give the same output, such as \((-2,4)\) and \((2,4)\). Only a repeated \(x\) with different \(y\)-values breaks the function.
What is the natural domain of a function?
The largest set of \(x\)-values for which the rule gives a real number. For \(\dfrac{1}{x-3}\) it is \(\mathbb{R}\setminus\{3\}\); for a polynomial it is all of \(\mathbb{R}\).