Question

the graph shows two translations of $y = \sqrt{x}$. for the function $y = \sqrt{x - 7} + 1$, the domain is $x \geq \boldsymbol{\text{dropdown}}$ and the range is $y \geq \boldsymbol{\text{dropdown}}$. for the function $y = \sqrt{x - 7} - 1$, the domain is $x \geq \boldsymbol{\text{dropdown}}$ and the range is $y \geq \boldsymbol{\text{dropdown}}$.

Explanation:

Step1: Recall domain/range of square root

For \( y = \sqrt{x} \), domain \( x \geq 0 \), range \( y \geq 0 \).

Step2: Analyze \( y = \sqrt{x - 7} + 1 \)

• Domain: Inside sqrt \( x - 7 \geq 0 \Rightarrow x \geq 7 \).
• Range: \( \sqrt{x - 7} \geq 0 \Rightarrow \sqrt{x - 7} + 1 \geq 1 \Rightarrow y \geq 1 \).

Step3: Analyze \( y = \sqrt{x - 7} - 1 \)

• Domain: \( x - 7 \geq 0 \Rightarrow x \geq 7 \).
• Range: \( \sqrt{x - 7} \geq 0 \Rightarrow \sqrt{x - 7} - 1 \geq -1 \Rightarrow y \geq -1 \).

Answer:

For \( y = \sqrt{x - 7} + 1 \): domain \( x \geq \boldsymbol{7} \), range \( y \geq \boldsymbol{1} \).
For \( y = \sqrt{x - 7} - 1 \): domain \( x \geq \boldsymbol{7} \), range \( y \geq \boldsymbol{-1} \).