Question

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.

Answer 1

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 

y³ = 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: