Question

How do you find the domain and range for $y=\dfrac{3(x-2)}{x}$ ?

Answer 1

In this question, we have to find the domain and range of an equation. As we know, domain means the set that contains the input of a function. Also, the range is the set that contains all the output the value of a function. Thus, for the domain, we will find where x is not defined in the given equation. And for range, we will find the value of x in terms of y by using the basic mathematical rules, which is the required solution to the problem.

Complete step by step answer:
According to the question, we have to find the domain and the range of a function.
The function given to us is $y=\dfrac{3(x-2)}{x}$ -------- (1)
Now, for finding the domain of equation (1), we see that it a fractional number.
As we know, the fractional number is a number expressed in the form of $\dfrac{p}{q}$ , where p and q are integers and $q\ne 0$ , therefore from the definition and equation (1), we get that
$x\ne 0$
Thus, the domain for the given function is any real number except 0, which we can express as
domain=R\sim \left\\{ 0 \right\\} ----- (2)
Now, for the range we will change the given equation in terms of y, that is
We will first multiply x on both sides in the equation (1), we get
$y.x=\dfrac{3(x-2)}{x}.x$
On further solving, we get
$yx=3(x-2)$
Now, we will apply the distributive property $a(b-c)=ab-ac$ in the above equal, we get
$yx=3x-6$
Now, we will subtract 3x on both sides in the above equation, we get
$yx-3x=3x-6-3x$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$yx-3x=-6$
Now, we will take x common from the left-hand side in the above equation, we get\
$x(y-3)=-6$
Now, we will divide (y-3) on both sides in the above equation, we get
$\dfrac{x(y-3)}{y-3}=\dfrac{-6}{y-3}$
On further simplification, we get
$x=\dfrac{-6}{y-3}$ ----- (3)
Therefore, as per the definition of fractional numbers, it is expressed in the form of $\dfrac{p}{q}$ , where p and q are integers and $q\ne 0$ , therefore from equation (3), we get
$y-3\ne 0$
Now, we will add 3 on both sides in the above equation, we get
$y-3+3\ne 0+3$
As we know, the same terms with opposite signs cancel out each other, thus we get
$y\ne +3$
Thus, the range for the given problem is any real number except 3, that is
range=R\sim \left\\{ 3 \right\\}

Therefore, for the equation $y=\dfrac{3(x-2)}{x}$ , its domain is equal to R\sim \left\\{ 0 \right\\} and its range is R\sim \left\\{ 3 \right\\}

Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical error. Do not forget the definition of fractional number, domain, and range.