Question

What is the domain and range of `y = sqrt(x-10) + 5`?

Answer 1

Domain: `[10, +oo)`
Range: `[5, +oo)`

Explanation:

Let's start with the domain of the function.

The only restriction you have will depend on `sqrt(x-10`. Since the square root of a number will produce a real value only if that number if positive, you need `x` to satisfy the condition

`sqrt(x-10)>=0`

which is equivalent to having

`x-10 >=0 => x>=10`

This means that any value of `x` that is smaller than `10` will be excluded from the function's domain.

As a result, the domain will be `[10, +oo)`.

The range of the function will depend on the minimum value of the square root. Since `x` cannot be smaller than `10`, `f(10` will be the starting point of the function's range.

`f(10) = sqrt(10-10) + 5 = 5`

For any `x>10`, `f(x)>5` because `sqrt(x-10)>0`.

Therefore, the range of the function is `[5, +oo)`

graph{sqrt(x-10) + 5 [-3.53, 24.95, -3.17, 11.07]}

SIDE NOTE Move the focus of the graph 5 points up and 10 points to the right of the origin to see function.