How to determine the range and domain of a function

Question

I'm practicing questions on domain and range and how it affects the graph whenever it is even or odd, I realized a relationship in some questions but I'm not certain on whether or not they are correct. My thoughts were that for all odd powers or roots, the domain is all real numbers. I'm unable to prove why because that's why it's plainly so, I visited some YouTube videos to help guide me but they never showed an actual reason and brushed over it for it to be left the way it is.

Basically, I'm trying to understand how the rules applied for domains and ranges are affected pertaining to the questions below. (This is not for an assignment, this is just my own personal study).

What is the domain and range of the functions below, outline with steps?

$$ f(x)=x^{1/4} \tag{1}$$

$$ f(x)=x^{1/5} \tag{2}$$

$$ f(x)=x^4 \tag{3}$$

$$ f(x)=x^5 \tag{4}$$

$$ f(x)=x^{-3} \tag{5}$$

$$ f(x)=x^{-1/3} \tag{6}$$

Answer 1

These problems depend a lot on the function you are studying. For the domain, always try to find the largest possible subset of $\mathbb{R}$ for which the expression makes sense.

For example, the function $f(x) = x^{1/4} = \sqrt[4]x$ cannot be evaluated when $x < 0$, thus the highest possible domain for which it can be defined is $\mathbb{R}_{\ge 0} = \{ x \in \mathbb{R} : x \ge 0\}$.

To determine the range, you can observe that the fourth root of a positive number will always be positive, so its range will be included in $\mathbb{R}_{\ge 0}$.

Alternative, given $x \in \mathbb{R}_{\ge 0}$, we have that $f(x^4) = x$, so every positive number can be obtained as the fourth root of another nmber, thus the range is $\mathbb{R}_{\ge 0}$.