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 characteristics of both functions.
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Function \( f(x) = \sqrt{x} \):
- This function is defined only for \( x \geq 0 \) (the square root of negative numbers is not real).
- It starts at the origin (0,0) and increases as \( x \) increases, approaching infinity as \( x \) approaches infinity.
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Function \( f(x) = \sqrt{x} + 3 \):
- This is a vertical shift of \( f(x) = \sqrt{x} \) upwards by 3 units.
- It is also defined only for \( x \geq 0 \) and starts at the point (0,3).
The key points for analysis:
- Both functions are only valid for \( x \geq 0 \).
- The second function (\( f(x) = \sqrt{x} + 3 \)) maintains the same shape as the first function but is shifted up by 3 units.
Evaluating the Graphs:
- We should look for a graph that shows:
- The function \( \sqrt{x} \) starting at (0,0) and increasing.
- The function \( \sqrt{x} + 3 \) starting at (0,3) and also increasing, but always 3 units above \( \sqrt{x} \).
Since both functions exist only for non-negative \( x \), the graph should not display points for negative x-values.
Given the descriptions of the graphs, you want to ensure you choose one that contains these elements. The graphs should show the two curves starting at the respective points and existing only in the first quadrant starting at x=0.
Among the options, one of them should depict both functions as described, looking for the indicated starting points and confirming the vertical shift.
My advice would be to pick the graph that displays these properties (if available). If there's ambiguity, please look closely to find visual indications of the vertical shift specifically around the points (0,0) and (0,3).
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