Question

over which interval is the graph of the parent absolute value function $f(x) = |x|$ decreasing?
$\bigcirc$ $(-\infty, \infty)$
$\bigcirc$ $(-\infty, 0)$
$\bigcirc$ $(-6, 0)$
$\bigcirc$ $(0, \infty)$

Explanation:

Step1: Recall the absolute value function

The parent absolute value function \( f(x) = |x| \) can be written as a piecewise function:

$$ f(x) =
LATEXBLOCK0
$$

Step2: Analyze the slope of each piece

• For \( x \geq 0 \), \( f(x)=x \), which has a slope of \( 1 \) (positive), so the function is increasing on \( (0, \infty) \).
• For \( x < 0 \), \( f(x)= -x \), which has a slope of \( -1 \) (negative), so the function is decreasing on \( (-\infty, 0) \).

Step3: Evaluate the options

• Option \( (-\infty, \infty) \): The function is not decreasing everywhere (it increases for \( x \geq 0 \)).
• Option \( (-\infty, 0) \): Matches the interval where the function is decreasing (slope is negative here).
• Option \( (-6, 0) \): This is a sub - interval of \( (-\infty, 0) \), but the question asks for the interval over which the graph is decreasing for the parent function. The parent function's decreasing interval is \( (-\infty, 0) \), not just \( (-6,0) \).
• Option \( (0, \infty) \): The function is increasing here, not decreasing.

Answer:

B. \((-\infty, 0)\)