Find the Arguments of a given Absolute-value function

Question

a) How can I find the arguments of the absolute-value function, y=|x-2|-x. b) How can I sketch the graph with vertex indicated with the intercepts. c) what will be the range of this funtion?

Answer 1

a)

Absolute value equation must be solved in two parts

1) Assume the argument (x - 2) is a positive number, i.e. x ≥ 2.

y = x - 2 - x

y = -2

2) Assume the argument (x - 2) is a negative number, i.e. x < 2

y = -(x - 2) - x

y = -x + 2 - x

y = -2x - 2

b)

One part of the graph (x ≥ 2) is y = -2

One part of the graph (x < 2) is y = -2x - 2

Both are linear from Algebra 1.

c)

Look at the graph.

Answer 2

If x ≥ 2, then y = (x - 2) - x = -2 (graph is the horizontal ray with endpoint (2,-2) and containing (3.-2))

If x < 2, then y = -(x - 2) - x = -2x + 2 (graph is the ray with endpoint (2,-2) and containing (0,2))

Range = [ -2, ∞)