Question

What is the domain and range of $f\left( x \right)=\dfrac{x+9}{x-3}?$

Answer 1

If we have a function $f$ from a set $A$ to another set $B,$ then we call the set $A$ from which the mapping happens the domain of the function and the set of elements in the set $B$ to which the mapping occurs the range of the function.

Complete step by step solution:
Let us consider the given function $f\left( x \right)=\dfrac{x+9}{x-3}.$
We are asked to find the domain and the range of the given function.
Let us first find the domain of the function.
We know that the domain of a function is the set of elements from which the values mapped to their images.
So, every element in the domain should be mapped to an element which is called the image of the element and no element can have more than one image.
Now, from the given function which is a polynomial fraction, we can say that the function is not defined when $x=3.$ Because, we know that a fraction is not defined when the denominator is zero, and when $x=3,\,x-3=0.$
So, we can say that the domain of the function includes ever real number except $3.$
Therefore, the domain is \mathbb{R}- \left\\{ 3 \right\\}.
Now, let us find the range of the function.
As we know, the range of the function is the set of images of the elements in the domain.
We know that when $x=3$ from the negative side, the function assumes the value $-\infty$ and when the function is $x=3$ from the positive side, the function assumes the value $\infty .$
Therefore, the range of the function is $\mathbb{R}.$
Hence the domain of the function is \mathbb{R}- \left\\{ 3 \right\\} and the range of the function is $\mathbb{R}.$

Note: We know that the set $B$ to which the function $f:A\to B$ maps is called the codomain of the function. It is possible that every element in $B$ is not an image of some elements in $A.$ So, when every element is an image, we say that the codomain is the range of the set. So, the range is either the subset of the codomain or both are the same set.