Question

what is the range of this function?
y = |x - 5| + 3
{y|y > 3}
all real numbers
{y|y ≥ 3}
{y|y ≥ 5}

Explanation:

Step1: Recall absolute value property

The absolute value of any real number \( x \), denoted \( |x| \), satisfies \( |x| \geq 0 \) for all real \( x \). For the expression \( |x - 5| \), let \( u=x - 5 \), so \( |u| \geq 0 \).

Step2: Analyze the function \( y = |x - 5|+3 \)

Since \( |x - 5| \geq 0 \), adding 3 to both sides of the inequality gives \( |x - 5|+3 \geq 0 + 3 \), which simplifies to \( y\geq3 \). So the range of the function \( y = |x - 5|+3 \) is all real numbers \( y \) such that \( y\geq3 \), or in set - builder notation \( \{y|y\geq3\} \).

Answer:

\(\{y|y \geq 3\}\)