Working with functional notation: evaluating a function f of x for a given input, finding the input that gives a required output, and identifying a function from its formula.

What this dot point is asking

The SQA wants you to read and use function notation: evaluate a function $f(x)$ for a given input (including negatives and fractions), find the input that produces a required output by forming and solving an equation, and substitute an expression into a function.

What f(x) means

The notation $f(x)$ is read "f of x" and names a function: a rule that takes an input $x$ and produces one output. The letter $f$ is just the name of the rule (other common names are $g$ and $h$), and whatever is written inside the brackets is the input that replaces $x$ in the formula.

So if $f(x) = x^2 + 1$, then $f(3)$ means "put $3$ in place of $x$": $f(3) = 3^2 + 1 = 10$. The notation is compact and exact, which is why it is used throughout the course.

Evaluating a function

To evaluate, substitute the given value for every $x$ in the formula and simplify. Negative and fractional inputs need care, especially inside powers.

Finding the input for a given output

If you know the output and want the input, set the formula equal to that output and solve the resulting equation.

Substituting an expression

Sometimes the input is itself an expression. Replace every $x$ with that whole expression, keeping it in brackets to avoid errors.

This idea of putting one expression in place of the input is the foundation of composite functions, which are studied at Higher.

Reading values from a graph

A function can also be given as a graph rather than a formula. To find $f(2)$ from a graph, go to $x = 2$ on the horizontal axis, read up to the curve, and across to the $y$ value: that height is $f(2)$. To solve $f(x) = 3$, do the reverse: find where the curve is at height $3$ and read off the $x$ value (or values). The same two questions, evaluating and solving, appear whether the function is a formula or a graph.

Two functions in one question

A question may define more than one function, such as $f$ and $g$, and ask you to combine their values. Work out each separately and then combine. For $f(x) = 2x$ and $g(x) = x + 5$, the value $f(3) + g(4)$ is $6 + 9 = 15$. Treat each function name as its own rule, and always read carefully which function and which input each part is asking about.

Examples in context

Function notation is how science and computing describe rules. A formula for the cost of hiring a tool, $C(d) = 8d + 15$ pounds for $d$ days, is a function: $C(3) = 8 \times 3 + 15 = \pounds 39$ gives the cost of a three-day hire. Asking "how many days does $\pounds 63$ buy" means solving $8d + 15 = 63$, giving $d = 6$. The same notation turns a real-world rule into something you can evaluate or invert.

Try this

Q1. Given $f(x) = 4x - 1$, find $f(3)$. [1 mark]

• Cue. $4 \times 3 - 1 = 11$.

Q2. Given $g(x) = x^2 + 2$, find $g(-3)$. [2 marks]

• Cue. $(-3)^2 + 2 = 11$.

Q3. Given $h(x) = 3x + 5$, find $x$ when $h(x) = 20$. [2 marks]

• Cue. $3x + 5 = 20$, so $x = 5$.