How do you find the domain and range of #y=-abs(x)+2#?

1 Answer
Oct 7, 2017

Answer:

Domain: #{x | x in RR}#
Range: #{y | y le 2}#

Explanation:

The domain is all numbers that #x# could be without making #y# undefined.

In this case, there isn't any value that will make #y# undefined, since there are no fractions with #x# in the denominator or functions with undefined values (#|x|# is defined for all real numbers). Therefore, the domain of this function is all real numbers, or #RR#.

The range is every value that #y# could be for the values of #x# in the domain.

We know that the range of #y = |x|# is #y ge 0#, since the absolute value function returns only positive numbers, or 0 if the input is 0.

This means that the range of #y = -|x|# is #y le 0#, since we're taking every value in the range and making it negative.

This means that the range of #y = -|x|+2# is #y le 2#, since we're adding #2# to every value in the range.

Therefore, the domain is #x in RR# and the range is #y le 2#.

This is what the graph of this function looks like (notice that all values of #x# have a point at some #y# value, and all values of #y# less than or equal to #2# have a point at some #x# value):
graph{y = -(abs(x))+2 [-10, 10, -5, 5]}