Quick questions on Functions, relations and notation for HSC Maths Advanced: the function machine and $f(x)$ notation, evaluating and substituting into a function, function versus relation, the vertical line test, and classifying relations as one-to-one, many-to-one, one-to-many or many-to-many with the horizontal line test

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is a function is a rule with one output per input?

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Informally,

y

is a function of

x

when

y

is completely determined by

x

through some rule. The variable

x

is the independent variable (the input you choose) and

y

is the dependent variable (the output the rule hands back). The single defining property is this: each input has exactly one output. A rule that could give two different outputs for the same input is not a function.

What are substituting expressions, not just numbers?

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The input to a function need not be a number: you can substitute another letter, or a whole expression. The rule is always the same, replace every

x

with whatever is in the brackets, then simplify. This skill is the engine of differentiation from first principles later, so it is worth drilling now.

What is the vertical line test?

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Because a function forbids one input from having two outputs, you can spot a function straight off its graph: two points with the same

x

lie on the same vertical line. So draw vertical lines and look for a double crossing.

What is the four types of relations, stage by stage?

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Putting the two tests together classifies every graph into exactly one of four types. The vertical test decides function or not; the horizontal test then decides "to-one" or "to-many". Building the four cases up one at a time makes the pattern click: each panel below changes one test result, sweeping through all four combinations.

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