Question

Which of the following represents the absolute value function as a piecewise function?

a) f(x) = x, x ≥ 0; f(x) = -x, x < 0
b) f(x) = x, x > 0; f(x) = -x, x ≤ 0
c) f(x) = x, x < 0; f(x) = -x, x ≥ 0
d) f(x) = x, x ≤ 0; f(x) = -x, x > 0

Answer 1

Option '(a) f(x) = x, x ≥ 0; f(x) = -x, x < 0' correctly represents the absolute value function as a piecewise function, as it mirrors the behavior of |x| by outputting x when x is non-negative and -x when x is negative. The correct answer is option a.

Step-by-step explanation:

To identify the correct representation of the absolute value function as a piecewise function, we need to focus on how the absolute value function operates. The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of whether x is positive or negative. Essentially, |x| equals x when x is non-negative (0 or positive) and -x when x is negative.

Now, considering the options provided, option '(a) f(x) = x, x ≥ 0; f(x) = -x, x < 0' exactly matches the behavior of the absolute function. When x is non-negative (greater than or equal to 0), the function is just x; when x is negative (less than 0), the function is the negation of x, or -x. Therefore, the piecewise function that correctly represents the absolute value function is:

f(x) = x, x ≥ 0; f(x) = -x, x < 0