# What is the domain and range of f(x)=abs(x) written in interval notation?

Apr 6, 2018

Domain: $\left(- \infty , \infty\right)$
Range: $\left[0 , \infty\right)$

#### Explanation:

The domain of a function is the set of all $x$ values that give a valid result. In other words, the domain consists of all the $x$ values you are allowed to plug into $f \left(x\right)$ without breaking any math rules. (Like dividing by zero.)

The range of a function is all the values that the function can possibly output. If you say that your range is $\left[5 , \infty\right)$, you are saying that your function cannot ever evaluate to less than 5, but it can certainly go as high as it wishes.

The function you give, $f \left(x\right) = | x |$, can accept any value for $x$. This is because every number has an absolute value. The absolute value of $5$ is $| 5 | = 5$. The absolute value of $- 3$ is $| - 3 | = 3$. Any number can be plugged in, so our domain is as large as possible, that is, $\left(- \infty , \infty\right)$.

Our range, however, is not so broad. All positive numbers stay positive. All negative numbers get turned into positive numbers. (Since this is what the absolute value operator does.) Thus, our function cannot output a negative number. So our range is $\left[0 , \infty\right)$.