For $$x \in \mathbb{R}$$, two real valued functions $$f(x)$$ and $$g(x)$$ are such that, $$g(x) = \sqrt{x} + 1$$ and $$fog(x) = x + 3 - \sqrt{x}$$. Then $$f(0)$$ is equal to

Given $$g(x) = \sqrt{x} + 1$$ and $$f \circ g(x) = x + 3 - \sqrt{x}$$.

Express $$f$$ in terms of a new variable.

Let $$t = g(x) = \sqrt{x} + 1$$, so $$\sqrt{x} = t - 1$$ and $$x = (t - 1)^2$$.

Find $$f(t)$$.

$$f(t) = f(g(x)) = x + 3 - \sqrt{x} = (t-1)^2 + 3 - (t-1)$$

$$= t^2 - 2t + 1 + 3 - t + 1 = t^2 - 3t + 5$$

Find $$f(0)$$.

$$f(0) = 0 - 0 + 5 = 5$$

The answer is Option B: $$5$$.

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