Question

What is the domain and range of $y={{x}^{3}}?$

Answer 1

Hint: We know that the domain of a function is the set of elements from which the values mapped to their images. And we also know that the range of a function is the set of elements to which the values of domain mapped.

Complete step by step solution:
Let us consider the given function $y={{x}^{3}}.$
We are asked to find the domain and the range of the given function.
Let us first discuss the domain of the function.
We know that the domain of a function is the set of elements from which the values mapped to their images.
So, every element in the domain should be mapped to an element which is called the image of the element and no element can have more than one image.
As we know, the given function is a polynomial function. And so, we can say that we can map every real number to a number which is its cube. No real number has two cubes. So, we can say that the set of real numbers is the domain of the given function.
Now, let us discuss the range of the function.
As we know, the range of the function is the set of images of the elements in the domain.
So, we can say the range of the given polynomial function is also the set of real numbers for which we can find a cube root for each real number.
Hence the domain and the range of the given polynomial function is $\mathbb{R}.$

Note: We know that the range set is also called the image set. For every element in the range set, there corresponds an element in the domain and this element is called the preimage of the element. The set of elements to which the function maps the elements of the domain is called the codomain. Also, remember that a polynomial function is always continuous.