Question

what are the domain and range of this function?
y = -2|x|
domain: {x | x < 0}, range: {y | y ≤ 0}
domain: all real numbers, range: all real numbers
domain: all real numbers, range: {y | y ≤ 0}
domain: {x | x < 0}, range: all real numbers

Explanation:

Step1: Analyze Domain of \( y = -2|x| \)

The function \( y = -2|x| \) involves the absolute value of \( x \). The absolute value function \( |x| \) is defined for all real numbers (since we can take the absolute value of any real number). Multiplying by \(-2\) doesn't restrict the input \( x \), so the domain (all possible \( x \)-values) is all real numbers. This eliminates options with domain \( \{x | x < 0\} \).

Step2: Analyze Range of \( y = -2|x| \)

Recall that \( |x| \geq 0 \) for all real \( x \). Multiply both sides by \(-2\) (remember to reverse the inequality sign when multiplying by a negative number): \( -2|x| \leq 0 \). So \( y = -2|x| \) means \( y \leq 0 \) for all real \( x \). Thus, the range (all possible \( y \)-values) is \( \{y | y \leq 0\} \). This eliminates options with range "all real numbers".

Answer:

C. domain: all real numbers, range: \( \{y | y \leq 0\} \) (assuming the third option is labeled C; if the options are unlabeled, the correct one is the third option in the list: "domain: all real numbers, range: \( \{y | y \leq 0\} \)")