Question

How do you find the domain and range of $2x + 3y = 12$?

Answer 1

Hint: Domain is the set of all elements for which the given function is well-defined. This can be found by using the well-defined property of functions. Range is the set of all outputs given by a function. This can be found by putting the various domain values in the function.

Complete step by step solution:
First, we will express the given equation in the form $y = f(x)$. And then will proceed in finding the domain and range of f.
The given equation is $2x + 3y = 12$.
By rearranging y term to one side,
$ \Rightarrow 3y = 12 - 2x$
Now, by dividing the whole expression by 12, we get
$y = 4 - \dfrac{2}{3}x$
$ \Rightarrow f(x) = y = 4 - \dfrac{2}{3}x$
Observe that the function $f(x)$ is a linear expression which is well defined for all values of $x \in \mathbb{R}$.
So, the domain of the function is the entire set of real numbers $\mathbb{R}$.
Also observe that the function $f(x)$ can give back any real number when an appropriate real number is taken as x.
So, the range of the function is also the entire set of real numbers $\mathbb{R}$.

Note:
Note that the domain of polynomial functions is always the entire real line $\mathbb{R}$. But their range may vary. Also, polynomial functions are continuous throughout the real line. We will use the well-defined property of a function to find the domain of a function. In case your function contains the variable inside the root, then the value inside the root can not be negative. Also, if the variable is in the denominator, then check that the denominator should not be equal to zero. These simple rules can help determine the domain of a function.