Question

How do you find the domain of the functions `root(4)( -4-7x)` and `root(5)( -4-7x)`?

Answer 1

Both of these function have radicand `(-4-7x)`, The difference is that in one case we are taking an even root (4th) and in the other, we want an odd root.

Even roots of negative numbers are not in the set of real numbers. (They are in the complex, or 'two-part' numbers.) Odd roots of negative numbers are negative numbers.

So, there is no restriction on the domain of `root(5)( -4-7x)`. The domain is all real numbers. This may also be written R or `(-oo, oo)`

For `root(4)( -4-7x)` , in order to avoind wandering into imaginary-land, we need to make sure that `-4-7x` is not negative.

That is, we need :`-4-7x >=0` . We solve this inequalipty:

`-4-7x >=0`
`-4 >=7x` , so

`7x <= -4` , and

`x <= -4/7` .

The domain is the solutions set for: `x <= -4/7` .

Perhaps your teacher prefers you to write: All real `x` less than or equal to `-4/7` or : `(-oo, -4/7)`