To determine which graph accurately demonstrates the relationship between the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \), we can analyze the functions:
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Function \( f(x) = \sqrt{x} \):
- This function is the square root function, which starts at the origin (0,0) and increases as \( x \) increases. It is defined only for \( x \geq 0 \), and its values range from 0 upwards.
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Function \( f(x) = \sqrt{x} + 3 \):
- This function is simply the square root function shifted up by 3 units. It starts at the point (0,3) and also increases as \( x \) increases.
Based on this analysis:
- The graph of \( f(x) = \sqrt{x} \) will begin at (0,0) and increase into the first quadrant.
- The graph of \( f(x) = \sqrt{x} + 3 \) will begin at (0,3) and similarly increase into the first quadrant.
Key Points for the Graph:
- The first curve (for \( f(x) = \sqrt{x} \)) should start from (0,0).
- The second curve (for \( f(x) = \sqrt{x} + 3 \)) should start from (0,3) and be parallel to the first curve.
All given options included both curves, but we need to check whether the y-values are accurately represented based on the calculated points (0, 0) for the first function and (0, 3) for the second function.
Based on this information, identify a graph that shows:
- One curve starting at (0,0) and one curve starting at (0,3).
- Ensure that the y-axis values accommodate the shift and that the lowest values shown at y = 0 or y = 3 accurately represent these functions.
If both graphs presented in the responses depicted these properties and only varied in the range of the y-axis, you might want to choose any that appropriately display the mentioned characteristics based on the y-axis range provided. The specific graph cannot be determined without visual representation, but look for the one that starts at (0,0) and (0,3) respectively.
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