Question

1. choose the correct graph of the function $y = \sqrt{x} + 1$

Explanation:

Step1: Analyze the domain of the function

The function is \( y=\sqrt{x}+1 \). For the square root function \( \sqrt{x} \), the domain is \( x\geq0 \) (in the set of real numbers). So the graph of \( y = \sqrt{x}+1 \) should only exist for \( x\geq0 \).

Step2: Analyze the \( y \) - intercept

To find the \( y \) - intercept, we set \( x = 0 \). Substitute \( x = 0 \) into the function: \( y=\sqrt{0}+1=1 \). So the graph should pass through the point \( (0,1) \).

Step3: Analyze the behavior of the function

The derivative of \( y=\sqrt{x}+1=x^{\frac{1}{2}} + 1 \) is \( y^\prime=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}} \). Since \( x>0 \) (in the domain where the function is differentiable), \( y^\prime>0 \), which means the function is increasing for \( x>0 \).

Step4: Compare with the given graphs

• The first graph: It has negative \( y \) - values and does not pass through \( (0,1) \) (it has a negative \( y \) - intercept), so it is incorrect.
• The second graph: It starts near \( x = 0 \), and as \( x \) increases, \( y \) increases (consistent with the derivative being positive), and it is likely that at \( x = 0 \), the \( y \) - value is around 1 (after considering the grid), so it matches the function \( y=\sqrt{x}+1 \).

Answer:

The Middle Graph (assuming the second graph is the middle one here, as there are two shown and likely a third not fully visible, but based on the function \( y = \sqrt{x}+1 \), let's analyze:

First, the domain of \( y=\sqrt{x}+1 \) is \( x\geq0 \) (since square root of negative numbers is not real in real - valued functions). So the graph should start at \( x = 0 \). When \( x = 0 \), \( y=\sqrt{0}+1=1 \). The function \( y = \sqrt{x} \) is a curve that starts at \( (0,0) \) and increases slowly, and \( y=\sqrt{x}+1 \) is a vertical shift up by 1 unit.

Looking at the first graph: It has negative \( y \) - values and seems to be defined for \( x\geq0 \) but has a negative \( y \) - intercept, which is wrong because at \( x = 0 \), \( y = 1 \).

Looking at the second graph: Let's check the \( y \) - value at \( x = 0 \). If we assume the graph starts near \( x = 0 \) with \( y\approx1 \) (after adjusting for the grid), and it is increasing (since the derivative of \( y=\sqrt{x}+1 \) is \( y^\prime=\frac{1}{2\sqrt{x}}>0 \) for \( x>0 \)), so the second graph (the lower one shown) is more likely. But since the user might have a set of three graphs, but based on the two shown, the second graph (the one with the curve starting near the origin and going up) is the correct one. So the answer is the second graph (the one below the first one in the image).