To find \( (g \circ f)(x) \), we need to evaluate the composite function, which means we first find \( f(x) \) and then substitute that into \( g(x) \).
Given:
- \( f(x) = \sqrt{x + 3} \)
- \( g(x) = \frac{x^2 + 2}{x} \)
Now, we calculate \( g(f(x)) \):
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First find \( f(x) \): \[ f(x) = \sqrt{x + 3} \]
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Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(\sqrt{x + 3}) = \frac{(\sqrt{x + 3})^2 + 2}{\sqrt{x + 3}} \]
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Simplify: \[ g(f(x)) = \frac{x + 3 + 2}{\sqrt{x + 3}} = \frac{x + 5}{\sqrt{x + 3}} \]
Thus, we have: \[ (g \circ f)(x) = \frac{x + 5}{\sqrt{x + 3}} \]
Now, let's match this with the given responses. The correct answer is:
\[ (g \circ f)(x) = \frac{x + 5}{\sqrt{x + 3}} \]
This corresponds to the second response you provided:
Response: \[ (g \circ f)(x) = \frac{x + 5}{\sqrt{x + 3}} \]