# The square root function · Year 12

> The square root function, Year 12: The domain of a root. Explanation, worked examples and free practice for the national curriculum in England.

In The square root function (Year 12) you practise one skill: The domain of a root.

The square root function is defined where the expression under the root is not negative, and its graph starts at the point where that expression is zero. The derivative of the square root of x is one over twice the square root of x; this rule is given, not proved. When there is an expression under the root, multiply by the derivative of that expression.

Find the tangent to a root graph as for any function: its gradient is the derivative at the point. In these questions the expression under the root is a perfect square at the point asked about, so the answers are exact fractions. To investigate a root function, start with the domain, and look for a turning point where the derivative is zero or at the end of the domain.

**Tip:** Always start with the domain: it tells you where the graph begins and which solution to reject.

## What you learn

### The domain of a root

- Example: Find the domain of the function f(x) = √(x − 6) .
- Answer: x ≥ 6
- Solution: The expression under the root must not be negative: x − 6 ≥ 0 x ≥ 6

The exercises are made by rules and the answers are worked out in code. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 12 topics](https://kixplo.com/uk/en/maths/year-12/)