Question:
Find the domain and the range of the cube root function,
f: $R \rightarrow R: f(x)=x^{1 / 3}$ for all $x \in R$. Also, draw its graph.
Solution:
Given:
$f(x)=x^{1 / 3} \forall x \in R$
To Find: Domain and range of the given function.
Here, $f(x)=x^{1 / 3}$
The domain of the above function would be,
Domain $(f)=(-\infty, \infty)\{x \mid x \in R\}$
Because all real numbers have a cube root. There is no value of x which makes the function undefined.
Now, to find the range
Consider $f(x)=y$
Then, $y=x^{1 / 3}$
$y^{3}=x$
Since f(x) is continuous, it follows that
Range $(f)=(-\infty, \infty)\{y \mid y \in R\}$
Because for every value of y there would be a cube of that value.
Graph:
