To determine which graph accurately demonstrates the relationship between the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \), we need to understand how these functions relate to each other.
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Function \( f(x) = \sqrt{x} \): The graph of this function starts from the point (0,0) and increases for \( x \geq 0 \). It is not defined for negative values of \( x \).
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Function \( f(x) = \sqrt{x} + 3 \): This is essentially the same as the first function but shifted upwards by 3 units. Therefore, it will start at the point (0,3) and increase for \( x \geq 0 \).
For a proper representation of both functions on the same graph, we need to ensure:
- Both curves start from their respective initial points (0,0) for \( f(x) = \sqrt{x} \) and (0,3) for \( f(x) = \sqrt{x} + 3 \).
- The y-axis must be set to accommodate at least the range from the minimum of \( \sqrt{x} \) at 0 (which is 0) to \( \sqrt{x} + 3 \) for \( x=0\) (which is 3), and more in the increasing values as \( x \) gets larger.
Given the descriptions provided, we would find that the option with a y-axis that runs from -6 to 6 would be able to display the range of both functions properly, allowing for the upward shift of \( 3 \) units in the second function relative to the first.
Therefore, the best options would be the graphs with ranges of y-values that accommodate both curves appropriately.
The correct response is:
A or D: "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."
All information indicates a similar description, but the one you should choose must address the proper upward shift and ensure the curve is within the expected range. Double-check which graph shows the features distinctly based on the relationship described above.