Question

How do you find the domain and range for $y=x$?

Answer 1

The given function gives a linear graph passing through the origin. The domain of a function can be found by considering which all values the $x$ can take up and if there’s any restriction given in the question. The question has no such restrictions, so $x$ can have any real number, that is, domain is $\mathbb{R}$ as $x\in \mathbb{R}$. Range of a function is the value or the output that is obtained in $y$ when values from the domain are put in $x$. Range is the set of real numbers $\mathbb{R}$ as $y\in \mathbb{R}$.

Complete step by step solution:
According to the given question, we have been given a function, whose domain and range is to be found.
Domain of a function refers to the permissible values that the independent variable can take up. It can be a set of real numbers, $\mathbb{R}$ or a set of natural numbers, $\mathbb{N}$ or even a set of integers, $\mathbb{Z}$.
Range of a function refers to the value obtained when the independent variable takes up value permissible to it. It can also be a set of real numbers, $\mathbb{R}$ or a set of natural numbers, $\mathbb{N}$ or even a set of integers, $\mathbb{Z}$.
The given function $y=x$ has no such restrictions so $x$ can take up any real values , therefore domain is the set of all real numbers, $\mathbb{R}$ as $x\in \mathbb{R}$ or we can also represent it as $(-\infty ,\infty )$.
Also, for range, the function $y=x$ gives a linear graph, so $y$ can also take up all the values taken by $x$. So, the range is the set of real numbers $\mathbb{R}$ as $y\in \mathbb{R}$ or we can represent this also as $(-\infty ,\infty )$.

Note: In this particular question we had no restriction on the value of $x$ and $y$, so we could take up any values. But if suppose there is a condition that $x>0$, then the domain will have no negative numbers and only positive real numbers or we can write it as $(0,\infty )$ and range will also be $(0,\infty )$.