To understand the relationship between the two functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \), we can analyze their graphs.
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Graph of \( f(x) = \sqrt{x} \):
- This graph starts at the point (0,0) and increases, approaching infinity as \( x \) increases. It is only defined for \( x \geq 0 \).
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Graph of \( f(x) = \sqrt{x} + 3 \):
- This graph is identical to the graph of \( f(x) = \sqrt{x} \), but it is translated vertically upwards by 3 units. This means it starts at the point (0,3) and also increases, approaching infinity as \( x \) increases.
Key Characteristics:
- The shape of both graphs will be the same (a half-parabola), but the second graph will be shifted vertically up by 3 units.
- For any value of \( x \), the value of \( f(x) = \sqrt{x} + 3 \) is always 3 units higher than \( f(x) = \sqrt{x} \).
If you were to visually compare the two graphs, you would see that:
- The graph of \( f(x) = \sqrt{x} \) is below \( f(x) = \sqrt{x} + 3 \) at all points.
- Both graphs will intersect the y-axis at \( (0, 0) \) for the first graph and \( (0, 3) \) for the second graph.
Therefore, the correct graph would show the first function starting from the origin and the second function starting from (0,3) and both looking like "half-Parabolas" that continue upwards to the right.