Question

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

Answer 1

Domain of function `y=|x|+2` is `-oo < x < +oo`.
Range is `2 <= y < +oo`.

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>=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
`-oo < x < +oo`

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