Which graph accurately demonstrates the relationship between the functions f(x)=x−−√

To accurately describe the relationship between the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \):

1. The function \( f(x) = \sqrt{x} \) is the standard square root function, which starts at the origin (0,0) and increases slowly as \( x \) increases. The range of this function is \( y \geq 0 \).

2. The second function \( f(x) = \sqrt{x} + 3 \) is simply the square root function shifted vertically upward by 3 units. Therefore, the new function will start at the point (0, 3) and also increases slowly from there. The range of this function is \( y \geq 3 \).

From the descriptions, the characteristics of these two functions should show that:

• The curve for \( f(x) = \sqrt{x} \) begins at (0,0) and stays below the one for \( f(x) = \sqrt{x} + 3 \), which starts at (0,3).

The possible graphs need to reflect this upward shift of 3 units for the second graph. It should be noted that both functions only exist for \( x ≥ 0 \) since the square root is not defined for negative \( x \).

None of the graphs described appear to give the correct range for \( f(x) = \sqrt{x} \) starting at (0,0) or for \( f(x) = \sqrt{x} + 3 \) starting at (0,3). The approximate y-ranges are also significant.

Based on this understanding, if you need to choose a graph, look for one where:

• One curve (the lower one) starts at (0, 0) and shapes to the right.
• The other curve (the upper one) starts at (0, 3) and also shapes to the right, remaining entirely above the first curve.

If you provide the corresponding graphs, I can help you choose the correct one.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Given the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \):

1. \( f(x) = \sqrt{x} \):

   • Starts at (0, 0) when \( x = 0 \) and is defined for \( x \geq 0 \).
   • As \( x \) increases, \( f(x) \) increases but remains non-negative.

2. \( f(x) = \sqrt{x} + 3 \):

   • Starts at (0, 3) when \( x = 0 \) and is also defined for \( x \geq 0 \).
   • This function is always 3 units higher than the first function.

Analyzing potential graphs:

• The first function \( f(x) = \sqrt{x} \) should be visible in the lower part of the graph starting from (0, 0).
• The second function \( f(x) = \sqrt{x} + 3 \) should be visible starting from (0, 3) and above the first curve.

Now, considering the provided graph options:

1. First option: Y-axis ranges from -4 to 4.
2. Second option: Y-axis ranges from -6 to 6.
3. Third option: Y-axis ranges from -6 to 6 (same as the second).
4. Fourth option: Y-axis ranges from -4 to 4 (same as the first).

Selection:

Both functions start at \( y \geq 0 \), so negative values on the y-axis in the first two options (-4 and -6) may render them less relevant unless those values are employed distinctly for the representation of the curves.

If we're focused on showing that \( f(x) = \sqrt{x} + 3 \) starts at (0, 3) and remains above \( f(x) = \sqrt{x} \), we should prioritize:

• The first option (Y-axis from -4 to 4) or the fourth option (also Y-axis from -4 to 4) could both show sufficient detail for the graphs of \( \sqrt{x} \) and \( \sqrt{x} + 3 \), but it is important that the curves remain non-negative.

Therefore, if I had to choose only one option, either the first or fourth option could potentially display the proper relationship. However, the crucial difference lies in ensuring that both curves do indeed appear correctly above zero on the y-axis where they should be. Without visual inspection, I would suggest going with the one where you can confirm this non-negativity validly.