Question

What is the domain and range of $f\left( x \right)=\dfrac{1}{x}$?

Answer 1

This question is from the topic of algebra. In this question, we will first know about the term domain. After that, we will know about the term range. After that, we will find out the domain of $f\left( x \right)=\dfrac{1}{x}$. After that, we will find out the range of $f\left( x \right)=\dfrac{1}{x}$. After that, we will get our answer.

Complete step by step answer:
Let us solve this question.
In this question, we have asked to find the domain and range of $f\left( x \right)=\dfrac{1}{x}$.
So, let us first know about the term domain.
The domain of any function (say the function of x that is,$f\left( x \right)$) is the set of all values for which the function is defined. The domain of any function cannot be that value or that set of values in which the value of that function is infinite.
A rational function is a function of the form $f\left( x \right)=\dfrac{p\left( x \right)}{q\left( x \right)}$, where $p\left( x \right)$ and $q\left( x \right)$ are polynomials and $q\left( x \right)\ne 0$. The domain of a rational function consists of all the real numbers $x$ except those for which the denominator is zero. To find the domain of any rational function, the values of $x$ will be excluded for those values in which the denominator equals zero.
The range of any function is the set of all values of any function that is, the domain of that function takes. Or, we can say that if we put all the values of domain in the function, then the resulting set of values gives us the range of that function.
So, in the function $f\left( x \right)=\dfrac{1}{x}$, we can say that at all values of $x$, the function $f\left( x \right)$ is defined except $x=0$. Because, at $x=0$ the value of the function $f\left( x \right)=\dfrac{1}{x}$ goes to infinity or undefined.
So, we can write the domain of function as \mathbb{R}-\left\\{ 0 \right\\}, where $\mathbb{R}$ is a set of all real numbers or we can write this as $\left( -\infty ,0 \right)\cup \left( 0,+\infty \right)$.
Now, for the range we can see that infinity is not in the domain, so after putting in the given function, then we will get the value as zero. So, we will consider zero in the range.
Now, we can say that after putting the values as $\left( -\infty ,0 \right)\cup \left( 0,+\infty \right)$ in the function $f\left( x \right)=\dfrac{1}{x}$, then we will get the range of function as $\left( -\infty ,0 \right)\cup \left( 0,+\infty \right)$ or \mathbb{R}-\left\\{ 0 \right\\}, where $\mathbb{R}$ is set of real numbers.
So, we get the domain as $\left( -\infty ,0 \right)\cup \left( 0,+\infty \right)$ and range as $\left( -\infty ,0 \right)\cup \left( 0,+\infty \right)$ of the function $f\left( x \right)=\dfrac{1}{x}$.

Note: We should have a better knowledge in the topic of algebra to solve this type of question easily. We should know about the terms domain and range. We should know how it is to find the domain of any function. We should know about how to find the range of any function to solve this type of question easily.