Find the domain and the range of the cube root function,

f: R → R: f(x) = x^{1/3} for all x ϵ R. Also, draw its graph.

Given:

f(x) = x^{1/3}∀ x є 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) = (-∞, ∞) {x | x є 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) = (-∞, ∞) {y | y є R}

Because for every value of y there would be a cube of that value.

Graph:

1