Question

what is the range of this function?
y = |x + 3| + 3
{y|y ≥ -3}
{y|y ≤ 3}
{y|y ≥ 3}
{y|y < -3}

Explanation:

Step1: Recall absolute value property

The absolute value of any real number \( x \), denoted \( |x| \), satisfies \( |x| \geq 0 \) for all real numbers \( x \). For the expression \( |x + 3| \), let \( u=x + 3 \). Then \( |u| \geq 0 \), so \( |x+3| \geq 0 \) for all real \( x \).

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

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

Answer:

\(\{y|y\geq3\}\)