Question

How do you find the domain and range of `f(x)=sqrt(x-4)`?

Answer 1

Here's what I got.

Explanation:

You know that when working with real numbers, you can only take the square root of a positive number.

This implies that the domain of the function, which includes all the values that `x` can take for which `f(x)` is defined, will have to account for the fact that

`x - 4 >= 0`

This is equivalent to saying that

`x >= 4`

You can thus say that the domain of this function is all real numbers that satisfy the above condition. In interval notation, this will be `x in [4, +oo)`.

The range of the function tells you the values that the function can take for values that `x` can take.

In this case, if you take the square root of a positive number, you will end up with a positive number, so

`f(x) = sqrt(x - 4) >= 0 color(white)(.)(AA) x in [4, +oo)`

The minimum value that `f(x)` can take occurs when `x = 4`, so

`f(4) = sqrt(4 - 4) = sqrt(0) = 0`

For any other value of `x >4`, you will have `f(x) >0`. In interval notation, the range of the function is `[0, + oo)`.

graph{sqrt(x-4) [-10, 10, -5, 5]}