Question

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

Answer 1

We are given with $y=\dfrac{3}{x}$ which means that it has a variable in the denominator. To compute the domain and range of a function with a variable in the denominator, we must set the denominator equal to zero and then we have to exclude $x$, we get an equation to be solved.

Complete step by step solution:
Now let us have a brief regarding the range and domain of functions. The domain means the set of possible input values. The graph of a domain consists of all the values that are shown upon the $x-axis$. The range is nothing but the set of possible output values. The graph of range consists of values that are represented upon the $y-axis$. We can find the domain and range by using graphs since both of them contain the required values that are to be plotted.
Now let us start finding the domain $y=\dfrac{3}{x}$.
While finding the domain, we shall not divide by $0$ because it will give undefined value of y.
Since $y$ is defined $\forall x\in R:x\ne 0$
R-\left\\{ 0 \right\\}
$\therefore$The domain of $y=\dfrac{3}{x}$ would be $\left( -\infty ,0 \right)\cup \left( 0,+\infty \right)$
Now let us find the range of $y=\dfrac{3}{x}$.
Let us consider that-

& \underset{x>0-}{\mathop{\lim }}\,y=-\infty \\\ & \underset{x<0+}{\mathop{\lim }}\,=+\infty \\\ \end{aligned}$$ $$\therefore $$ The range of $$y=\dfrac{3}{x}$$ is $$\left( -\infty ,+\infty \right)$$ **Note:** The rule of the domain is that if a function contains a square root, we must set the equation inside the square root greater or equal to zero and solve. The resulting answer would be the domain. If the function contains a fraction, set the denominator not equal to zero. By solving this, we obtain the domain. The graph of $$y=\dfrac{3}{x}$$ is