# The square root function · Grade 10 · 4 units

> The square root function, Grade 10 · 4 units: The domain of a root and more. Explanation, worked examples and free practice following the Israeli curriculum.

In The square root function (Grade 10 · 4 units) you practise 5 skills: The domain of a root · The derivative of a root · The tangent to a root graph · Investigating a root function · A parameter in a root function.

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 derivative of a root

- Example: Here is the function f(x) = √(x − 5) . Find f′(6) .
- Answer: 1/2
- Solution: f′(x) = 1 ÷ (2√(x − 5)) At x = 6 the expression under the root is 1 : f′(6) = 1 ÷ (2 × 1) = 1/2

### The tangent to a root graph

- Example: Find the gradient of the tangent to the graph of the function f(x) = √x at the point where x = 25 .
- Answer: 1/10
- Solution: f′(x) = 1 ÷ (2√x) m = f′(25) = 1 ÷ (2√25) = 1 ÷ (2 × 5) = 1/10

### Investigating a root function

- Example: At which point does the graph of the function f(x) = 3 − √(x + 3) start?
- Answer: (−3, 3)
- Solution: The function is defined for x ≥ −3 . The graph starts at the end of the domain: f(−3) = 3 . The point is (−3, 3)

### A parameter in a root function

- Example: Here is the function f(x) = √(kx − 1) . You also know that f(5) = 2 . Find k .
- Answer: 1
- Solution: √(5k − 1) = 2 Square both sides: 5k − 1 = 4 5k = 5 k = 1

## Not on the site yet

- Sketching the graph
- Typing a derivative with a root
- The root of a polynomial of degree above two

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 4 units topics](https://kixplo.com/il/en/math/grade-10-4u/)