Question

which of the following is the graph of this square root function? f(x) = \sqrt{x + 1} + 5

Explanation:

Step1: Find the domain of the function

For the square root function \( f(x)=\sqrt{x + 1}+5 \), the expression inside the square root must be non - negative. So we solve the inequality \( x + 1\geq0 \), which gives \( x\geq - 1 \). This means the graph of the function should start at \( x=-1 \) (or to the right of \( x = - 1 \)).

Step2: Find the y - intercept (when \( x = 0 \))

Substitute \( x = 0 \) into the function \( f(x)=\sqrt{x + 1}+5 \). We get \( f(0)=\sqrt{0 + 1}+5=1 + 5=6 \).

Step3: Analyze the starting point (when \( x=-1 \))

Substitute \( x=-1 \) into the function \( f(x)=\sqrt{x + 1}+5 \). We have \( f(-1)=\sqrt{-1 + 1}+5=0 + 5 = 5 \). So the graph should pass through the point \( (-1,5) \).

Now let's analyze the three graphs:

• The first graph: When \( x=-1 \), we can check the y - value. And when \( x = 0 \), \( y = 6 \), and the domain starts at \( x=-1 \), which matches our calculations.
• The second graph: The domain seems to start at \( x\lt - 1 \) (since the graph starts at \( x\) around - 4), which does not match the domain \( x\geq - 1 \) of our function.
• The third graph: When \( x=-1 \), let's check the y - value. If we assume the graph is drawn correctly, it does not pass through \( (-1,5) \) (the y - value at \( x=-1 \) is not 5).

Answer:

The first graph (the one on the left) is the graph of the function \( f(x)=\sqrt{x + 1}+5 \).