6.2Functions: One Input, One Output

A function is a rule that turns each input $x$ into exactly one output $y$ , written $y = f(x)$ . Press the same button on a vending machine and you always get the same one item, never two, never none.

The rule

f(x) = 2x + 1

drawn as a machine: an input enters on the left, each stage acts in turn, and exactly one output leaves on the right.

Each input must give exactly one output; if one input could give two, it is not a function.
$f(3)$ means substitute $x = 3$ into the rule to find the output.

Evaluating a function: whole number input

A function is defined by $\displaystyle f(x) = 4x - 7$ . Find $\displaystyle f(5)$ and $\displaystyle f(0)$ .

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\displaystyle f(5)

\displaystyle = 4(5) - 7 \quad \scriptsize\textit{substitute } x = 5

\displaystyle = 20 - 7

\displaystyle = 13

\displaystyle f(0)

\displaystyle = 4(0) - 7 \quad \scriptsize\textit{substitute } x = 0

\displaystyle = 0 - 7

\displaystyle = -7

Now you try

Given $y = 2x - 3$ , find the value of $y$ when $x = 1$ .

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\displaystyle y

\displaystyle = 2x - 3

\displaystyle y

\displaystyle = 2(1) - 3 \quad \scriptsize\textit{substitute x = 1}

\displaystyle = 2 - 3

\displaystyle = -1

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Evaluating a function: negative input

Given $\displaystyle g(x) = 3 - 5x$ , find $\displaystyle g(-4)$ and find the value of $\displaystyle x$ when $\displaystyle g(x) = 28$ .

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\displaystyle g(-4)

\displaystyle = 3 - 5(-4) \quad \scriptsize\textit{substitute } x = -4

\displaystyle = 3 + 20

\displaystyle = 23

To find

x

when

g(x) = 28

\displaystyle 3 - 5x

\displaystyle = 28

\displaystyle -5x

\displaystyle = 25 \quad \scriptsize\textit{subtract 3 from both sides}

\displaystyle x

\displaystyle = -5

Now you try

Given $y = -2x - 2$ , find the value of $x$ when $y = -8$ .

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\displaystyle y

\displaystyle = -2x - 2

\displaystyle -8

\displaystyle = -2x - 2 \quad \scriptsize\textit{substitute y = -8}

\displaystyle -2x

\displaystyle = -8 + 2

\displaystyle -2x

\displaystyle = -6

\displaystyle x

\displaystyle = \frac{6}{2}

\displaystyle = 3

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Predict

If $f(x) = x^2 - 3$ , what is $f(2)$ ? Decide before reading on.

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f(2)

means: substitute

x = 2

into the rule.

\displaystyle f(2)

\displaystyle = 2^2 - 3

\displaystyle = 1

It does not mean

f \times 2

: the letter names the rule, and the bracket holds its input.