What a Function Really Is — Grade 10 Maths (CAPS)

The idea underneath every graph you will sketch this year — one input, exactly one output. The four ways to show a function and how to move between them, function notation f(x), domain and range, the vertical line test, and how to find a rule from a table.

Also known as: function notation · f(x) · domain and range · vertical line test · input output · relationships between variables

Before you sketch a single parabola, you need the idea underneath it. A function is a rule that takes an input and gives back exactly one output. That word — exactly one — is the whole definition.

Put 3 into the rule “double it and add one” and you get 7. Not sometimes 7 and sometimes 8. Every time, 7. A rule that could give two different answers for the same input is a relationship, but it is not a function.

Each input is joined to exactly one output. That is what makes this a function.

1Four ways to say the same thing

The same function can be written four ways, and the ATP expects you to move between them in any direction.

In words: “multiply the input by 2, then add 1”
As a table: x: 1, 2, 3, 4 → y: 3, 5, 7, 9
As a formula: y = 2x + 1, or in function notation f(x) = 2x + 1
As a graph: the straight line through (1 ; 3), (2 ; 5), (3 ; 7) and (4 ; 9)

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A question will hand you one representation and ask for another. Words → formula, table → formula and formula → graph are the three that come up most.

2From a table to a formula

This is the one worth drilling. Look at what happens to y as x goes up by 1.

Worked ExampleFind the rule for the table x: 1, 2, 3, 4 → y: 3, 5, 7, 9

1
As x increases by 1, y increases by 2 each time.
A constant difference means the rule is linear: y = 2x + c.
2
Substitute a known pair, x = 1 and y = 3: 3 = 2(1) + c
Any row of the table will do.
3
c = 1
4
y = 2x + 1
5
Check with a DIFFERENT row: x = 4 gives 2(4) + 1 = 9 ✓
Always test against a row you did not use to build the rule.

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Finding the constant difference and writing y = 2x, forgetting the c. The difference gives you the coefficient of x only. You still have to substitute a point to find where the rule starts.

3Function notation: f(x)

Writing f(x) = 2x + 1 instead of y = 2x + 1 lets you name the input you are asking about.

f(3) means “the output when the input is 3” = 2(3) + 1 = 7

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Reading f(x) as f × x. The brackets are not multiplication — f is the name of the rule and x is what you feed it. f(3) is a single number, the answer you get from the rule.

Worked ExampleIf f(x) = x² − 4x, determine f(5), f(−2), and the value of x for which f(x) = 0.

1
f(5) = 5² − 4(5) = 25 − 20 = 5
Substitute 5 wherever x appears.
2
f(−2) = (−2)² − 4(−2) = 4 + 8 = 12
Keep the brackets on a negative input — (−2)² is 4, not −4.
3
f(x) = 0 → x² − 4x = 0 → x(x − 4) = 0
Working backwards means solving an equation.
4
x = 0 or x = 4
Two inputs can share an output — that is allowed. Two OUTPUTS for one input is not.

4Domain and range

The domain is the set of values you are allowed to put IN — all the x-values.
The range is the set of values that come OUT — all the y-values.

Read them off a graph by projecting: the domain is what the graph covers left-to-right, the range is what it covers top-to-bottom. For a straight line like y = 2x + 1 both are all real numbers; for y = x² the domain is all real numbers but the range is y ≥ 0, because squaring never gives a negative.

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Two inputs may share an output — f(−3) and f(3) are both 9 for f(x) = x², and that is a perfectly good function. The rule only forbids one input having two outputs.

5The vertical line test

Given a graph, there is a one-second check. Slide a vertical line across it. If the line ever crosses the graph twice, that x-value has two y-values — and it is not a function.

The dashed vertical line meets this curve twice, so one input has two outputs. It fails the test.

A non-vertical straight line, a parabola and an exponential graph all pass. A circle fails, and so does a parabola lying on its side.

6Where this is going

Everything else in the Functions topic is this idea with a specific rule attached: the straight line, the parabola, the hyperbola and the exponential graph. When a later question asks for the domain, the range, or the value of y when x = 3, it is asking the questions on this page.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Explain, in your own words, what makes a relationship a FUNCTION.
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Every input has exactly ONE output ✓; no input may give two different answers ✓
2
Write the rule “multiply by 3, then subtract 4” as a formula in function notation.
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f(x) = 3x − 4 ✓✓
3
Determine the rule for the table x: 1, 2, 3, 4 → y: 5, 8, 11, 14.
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Constant difference of 3, so y = 3x + c ✓; 5 = 3(1) + c so c = 2 ✓; y = 3x + 2 ✓
4
If f(x) = 4x − 7, determine f(3).
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f(3) = 4(3) − 7 ✓ = 5 ✓
5
If f(x) = x² + 2x, determine f(−3).
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(−3)² + 2(−3) = 9 − 6 ✓ = 3 ✓
6
If f(x) = 5x + 1, determine the value of x for which f(x) = 26.
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5x + 1 = 26 ✓; x = 5 ✓
7
Write down the domain and the range of f(x) = x².
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Domain: x ∈ ℝ (all real numbers) ✓; Range: y ≥ 0 ✓
8
Explain why f(x) = x² is still a function even though f(−4) and f(4) are both 16.
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Two different inputs are allowed to give the same output ✓; the rule is only broken if ONE input gives two different outputs ✓
9
State the vertical line test and say what it tells you.
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Slide a vertical line across the graph ✓; if it ever cuts the graph more than once the graph is not a function, because that x-value has more than one y-value ✓
10
A taxi charges R12 to get in plus R7 per kilometre. Write a formula for the cost C of a journey of k kilometres, and calculate the cost of a 15 km trip.
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C = 7k + 12 ✓✓; C = 7(15) + 12 = R117 ✓