Question

the graph shows a square root function. use what you know about domain to select all of the following functions that could be the one graphed. $f(x) = \sqrt{x} - 3$; $f(x) = \sqrt{x} - 1$; $f(x) = \sqrt{x + 1}$; $f(x) = \sqrt{3x - 3}$

Explanation:

Step1: Recall domain of square root function

The domain of a square root function \( f(x) = \sqrt{u} \) is \( u \geq 0 \), where \( u \) is the expression inside the square root. We need to find the domain of each given function and check if it matches the graph (the graph starts at \( x = 0 \) or \( x \) value such that the inside of the square root is non - negative at \( x = 0 \) or other relevant points).

Step2: Analyze \( f(x)=\sqrt{x}-3 \)

For \( f(x)=\sqrt{x}-3 \), the expression inside the square root is \( x \). The domain is \( x\geq0 \).

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

For \( f(x)=\sqrt{x}-1 \), the expression inside the square root is \( x \). The domain is \( x\geq0 \).

Step4: Analyze \( f(x)=\sqrt{x + 1} \)

For \( f(x)=\sqrt{x + 1} \), we set \( x+1\geq0\), which gives \( x\geq - 1 \). The domain starts at \( x=-1 \), not \( x = 0 \) as the graph seems to start at \( x = 0 \) (from the visual, the graph is near the y - axis at \( x = 0 \)).

Step5: Analyze \( f(x)=\sqrt{3x-3} \)

For \( f(x)=\sqrt{3x - 3} \), we set \( 3x-3\geq0\). Solving \( 3x-3\geq0\) gives \( 3x\geq3\) or \( x\geq1 \). The domain starts at \( x = 1 \), not \( x=0 \).

Now, looking at the graph, the square root function starts at \( x = 0 \) (since it is near the y - axis at \( x = 0 \)). The functions \( f(x)=\sqrt{x}-3 \) and \( f(x)=\sqrt{x}-1 \) have a domain starting at \( x = 0 \) (because the expression inside the square root is \( x \), so when \( x = 0 \), the inside of the square root is \( 0\geq0 \)). The function \( f(x)=\sqrt{x + 1} \) has a domain starting at \( x=-1 \) and \( f(x)=\sqrt{3x - 3} \) has a domain starting at \( x = 1 \), which does not match the graph's starting point (visually, the graph starts at \( x = 0 \)).

Answer:

\( \boldsymbol{f(x)=\sqrt{x}-3} \), \( \boldsymbol{f(x)=\sqrt{x}-1} \)