Question

what is the range of this function?
y = 5|x - 7|
all real numbers
{y|y ≤ 0}
{y|y ≥ 7}
{y|y ≥ 0}

Explanation:

Step1: Recall absolute value property

The absolute value \(|a|\) of any real number \(a\) satisfies \(|a| \geq 0\) for all real \(a\). So for \(|x - 7|\), we have \(|x - 7| \geq 0\).

Step2: Analyze the function \(y = 5|x - 7|\)

Multiply both sides of \(|x - 7| \geq 0\) by 5 (since 5 is positive, the inequality direction remains the same). We get \(5|x - 7| \geq 5\times0 = 0\). Since \(y = 5|x - 7|\), this means \(y \geq 0\). So the range of the function is all real numbers \(y\) such that \(y \geq 0\), which is \(\{y|y \geq 0\}\).

Answer:

\(\{y|y \geq 0\}\)