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

1 Answer
Jul 25, 2016

Answer:

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#.