Algebra · step 13 of 13

Function notation, domain & reading a graph

f(x) names the output of f at input x — it is not multiplication, and f(x + 1) means substitute x + 1 everywhere, not add 1 to the answer. The natural domain is every input where the formula makes sense, so you exclude zeros of denominators and negatives under even roots. The vertical line test decides whether a curve is a function at all.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Substitute carefullyf(x+1) replaces every occurrence of x with the whole expression (x+1), in brackets.
Find the natural domainExclude values making a denominator zero or an even root negative, and combine the restrictions.
Read the range from the graphThe range is the set of y-values attained. Look for turning points and asymptotes.
Apply the vertical line testA curve is the graph of a function iff no vertical line meets it more than once — a circle fails this.

Worked example

For f(x) = √(x − 2)/(x − 5), find the natural domain and compute f(6) and f(a + 2).

Root condition: x − 2 ≥ 0, so x ≥ 2. Denominator condition: x ≠ 5.
Domain: [2, 5) ∪ (5, ∞).
f(6) = √4 / 1 = 2.
f(a+2) = √((a+2) − 2)/((a+2) − 5) = √a/(a − 3).

Answer. Domain [2,5) ∪ (5,∞); f(6) = 2; f(a+2) = √a/(a − 3).

Where marks get dropped

These are the specific errors that cost credit on function notation, domain & reading a graph questions — QED's rubric penalises each of them separately.

Reading f(x) as f times x. It is an application, and the distinction matters as soon as you meet composition.
Computing f(x + 1) as f(x) + 1. Substitution replaces the variable, it does not shift the output.
Giving the domain as a single interval when an excluded point splits it. Use a union.

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Function notation, domain & reading a graph — frequently asked questions

What is the natural domain?

The largest set of inputs for which the formula produces a real number, when no domain is stated explicitly. Exclusions come from division by zero and even roots of negatives.

How do I find the range?

Sketch the graph, or solve y = f(x) for x and see which y admit a solution in the domain. Turning points and asymptotes bound it.

What does the horizontal line test tell me?

Whether the function is injective, hence invertible. A horizontal line meeting the graph twice means two inputs share an output.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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