Question

How do you find the domain and range of $2\left( {x - 3} \right)$ ?

Answer 1

Hint: In this question, we have been asked to find the domain and range of $2\left( {x - 3} \right)$. First, study what domain and range are. Then, look for the possible values of $x$. These possible values constitute our domain. The resultant values of $y$ that we get, after putting the various values of $x$ is our range. This will be our answer.

Complete step-by-step answer:
We have to find the domain and range of $2\left( {x - 3} \right)$. But first, let us find out the meaning of domain and range.
What is domain? In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. In simple words, domain is a set of all the possible values that can be put in the given equation or expression. These are the values of $x$.
What is range? Range is the set of all those values that a function takes when there is an input in the question. These are the values of $y$.
Now, let us move towards the question. First, we will find the domain.
What all can we put in the place of $x$? All the real numbers can take a place of $x$ as no number will give it an undefined value. In such linear functions, there is no value of $x$ which can make it undefined. Hence, the domain of the given equation is $\left( { - \infty ,\infty } \right)$.
Now, let us find the range. For this, I will plot the equation on the graph. The blue line is the graphical representation of $2\left( {x - 3} \right)$.

We can see in the graph that for every value of $x$, a certain value for $y$ also exists. Therefore, the range of the given function is $\left( { - \infty ,\infty } \right)$.

Note:
Besides range and domain, there is also a term called ‘Co-domain’.
What is Co-domain? Co-domain is a set of all the possible values of $y$. The possible values of $x$ is called domain and the possible values of $y$ is called co-domain, whereas the actual values of $y$ is called range. This is the difference between range and co-domain. Co-domain has all the possible values, whereas range has all the actual values.