How do you solve this type of math problem?

Question

I have to graph an absolute value function, and state it's domain and range. I forgot how to solve them. Here's an example, but I need you to tell me how to do it because there are 5 more I have to do.

y= -|x+1| -2

Answer 1

There are two parts to the graph; one where the term inside

the absolute value sign is ≥ 0 and the other is ≤ 0.

1. Let x + 1 ≥ 0. Therefore, x ≥ -1,

and y = -(x + 1) - 2, or, y = -x - 3.

2. Let x + 1 ≤ 0. Therefore, x ≤ -1,

and y = -[-(x + 1)] - 2, or, y = x - 1.

So now you plot the graph of y = -x - 3 for x ≥ -1,

then you plot the graph of y = x - 1 for x ≤ -1.

For y = -x - 3, the greatest value of y is when x = -1,

that is, y = -(-1) - 3 = -2.

Thus, the domain for y = -x - 3 is x ≥ -1,

and the range is y ≤ -2.

For y = x - 1, the greatest value of y is also when x = -1,

that is, y = -1 - 1 = -2.

Thus, the domain for y = x - 1 is x ≤ -1,

and the range is y ≤ -2.

For the graph as a whole, the domain is all x

and the range is y ≤ -2.