Question

what is the range of the function $g(x) = |x - 12| - 2$?
$\\{y \mid y > -2\\}$
$\\{y \mid y \geq -2\\}$
$\\{y \mid y > 12\\}$
$\\{y \mid y \geq 12\\}$

Explanation:

Step1: Recall the range of absolute value function

The absolute value function \(|a|\) has a range of \(|a| \geq 0\) for any real number \(a\). In the function \(g(x)=|x - 12|-2\), let \(a=x - 12\). So we have \(|x - 12|\geq0\).

Step2: Find the range of \(g(x)\)

To find the range of \(g(x)\), we start with the inequality for \(|x - 12|\). Add \(- 2\) to both sides of the inequality \(|x - 12|\geq0\).
We get \(|x - 12|-2\geq0 - 2\), which simplifies to \(g(x)\geq - 2\). In set - builder notation, the range of \(g(x)\) is \(\{y|y\geq - 2\}\).

Answer:

\(\{y|y\geq - 2\}\) (the second option)