Question

23)
a function is shown.
$f(x) = |x - 7| - 11$
what is the range of the function?
${y: y in mathbb{r}}$
${y: -11 < y < 7}$
${y: -11 < y}$
${y: y < -11}$

Explanation:

Step1: Recall the property of absolute value

The absolute value function \(|a|\) has a range of \(|a| \geq 0\) for all real numbers \(a\). In the function \(f(x)=|x - 7|-11\), let \(a=x - 7\), so \(|x - 7|\geq0\) for all real \(x\).

Step2: Analyze the range of \(f(x)\)

Since \(|x - 7|\geq0\), we can add \(- 11\) to both sides of the inequality. When we add \(-11\) to \(|x - 7|\), we get \(|x - 7|-11\geq0 - 11\), which simplifies to \(f(x)\geq - 11\). In set - builder notation, the range of the function \(y = f(x)\) is \(\{y:y\geq - 11\}\) (or \(\{y:-11\leq y\}\), but looking at the options, the closest is \(\{y:-11 < y\}\) considering possible notation differences, and the other options are incorrect: \(\{y:y\in R\}\) is wrong as the function has a lower bound, \(\{y:-11 < y < 7\}\) is wrong as there is no upper bound like 7, and \(\{y:y < - 11\}\) is the opposite of the correct range).

Answer:

\(\{y:-11 < y\}\)