# How do you find the domain and range of y=absx +2?

Jul 25, 2016

Domain of function $y = | x | + 2$ is $- \infty < x < + \infty$.
Range is $2 \le y < + \infty$.

#### Explanation:

Let's start from the definition of an absolute value of any real number.
For positive number or zero its absolute value is the same as a number itself.
For negative number its absolute value is its negation (that is a corresponding positive number).

In short,
$| R | = R$ for $R \ge 0$ and
$| R | = - R$ for $R < 0$.

Using this definition, we see that absolute value is defined for all real numbers, which means that the domain of function $y = | x | + 2$ is
$- \infty < x < + \infty$

According to definition of absolute value, $| x | \ge 0$ for all real numbers $x$ with $| x | = 0$ only for $x = 0$.
As $x$ increases to $+ \infty$ or decreases to $- \infty$, $| x |$ is increasing to $+ \infty$.
From this follows that $2 \le | x | + 2 < + \infty$ - that is the range of our function $y = | x | + 2$.