Question

How do you find the domain and range of an absolute value function?

Answer 1

First of all we will need to do is to find the set of values of x for which we will get a possible of y. This is known as the domain of a function. Secondly, to find the range, we will find all the possible values of y which we will get after putting the values of x. This will be our range.

Complete step by step answer:
Before starting the question, we must know what is the meaning of domain and range of a function.
Domain: It is the set of possible values of a function’s input. This basically means domain is the set of values we can give input to a function for which it can give us a value.For example, the domain of $x + 1$ is $\left( { - \infty ,\infty } \right)$as for every possible value of x, there will be a possible value of y. But for $\dfrac{1}{{x + 1}}$, our domain would be $\left( { - \infty , - 1} \right) \cup \left( { - 1,\infty } \right)$. This means, all the possible values except 1 since if we put x=-1, then the function will become $\dfrac{1}{0}$ which is not defined.

Range: Similarly, range is the set of possible values of a function’s output. This basically means, range is the set of values, the function can give as a value of y. For example if $y = \dfrac{1}{x}$where, x is an integer. Codomain of this function will be $y = |x + 1|$. As for any value we put in x, y will always be less than or equal to 1. And only for x=1, it will give y=1. So, now, heading onto the question absolute value functions are written something like $y = |x + 1|$

Now, to find the domain:-
Case 1: If you don’t have a denominator, then the domain will be $\left( { - \infty ,\infty } \right)$.
Case 2: If you have a denominator, then the domain will be all values other than those where the denominator will become 0.
Case 2: If it has a root in the denominator, the domain will be all values except the ones where the value is inside root<0.

To find the range:-
Case 1: If you don’t have a denominator, range will be $[0,\infty )$. This is because whenever the value of y will tend to be negative, the modulus sign will make it positive.
Case 2: If you have a denominator, then the range will be all values above 0 except the points where the denominator is equal to 0.
Case 3: If it has a root in the denominator, the domain will be all values more than 0 except the ones where the value is inside root<0.

Note: Codomain and range are similar words. Both mean the same, that is all the possible values of y that we get after putting all the values of x. Whereas domain is the opposite. All the values of x we can put to get a possible value of y.