Question

2. Define the absolute value function, y = |x|, as a piecewise function. Please include complete sentences and examples to justify your answer to receive credit.

Answer 1

y = |x| = x if x ≥ 0, -x if x < 0

absolute value can be interpreted as a function that does not allow negative real numbers, forcing them to be positive (leaving 0 alone). if the input x is more than or equal 0, then x stays positive so there is no need to do anything: "x if x ≥ 0".
if the input is less than 0, then it is an negative number and needs a negative coefficient to negate the negative: "-x if x < 0"

example: if x = -3, then it will take the "-x if x < 0" piece resulting in y = -(-3) = 3, which is what |-3| does

if x = 1, it will take the "x if x ≥ 0" piece and just have y = 1 which is what |1| does.

for x = 0, it will take the "x if x ≥ 0" and just have y = 0 which is what |0| does