To evaluate the composite function $(g \circ f)(\sqrt{3})$, we'll break it down step by step.

Step 1: Understand the composite function

The composite function $(g \circ f)(x)$ means that we first apply the function $f(x)$, and then use the result of $f(x)$ as the input to the function $g(x)$. Mathematically, this can be written as: $(g \circ f)(x) = g(f(x))$

Step 2: Apply $f(x)$ to $\sqrt{3}$

Given that $f(x) = 2x + 3$, let's substitute $\sqrt{3}$ into $f(x)$: $f(\sqrt{3}) = 2(\sqrt{3}) + 3 = 2\sqrt{3} + 3$

Step 3: Use the result in $g(x)$

Now, substitute $f(\sqrt{3}) = 2\sqrt{3} + 3$ into $g(x)$. The function $g(x) = x^2 - 4x$, so: $g(2\sqrt{3} + 3) = (2\sqrt{3} + 3)^2 - 4(2\sqrt{3} + 3)$

Step 4: Simplify the expression

We'll first expand both terms:

Expand $(2\sqrt{3} + 3)^2$:

$(2\sqrt{3} + 3)^2 = (2\sqrt{3})^2 + 2(2\sqrt{3})(3) + 3^2 = 12 + 12\sqrt{3} + 9 = 21 + 12\sqrt{3}$

Expand $-4(2\sqrt{3} + 3)$:

$-4(2\sqrt{3} + 3) = -8\sqrt{3} - 12$

Step 5: Combine terms

Now combine the two results: $g(2\sqrt{3} + 3) = (21 + 12\sqrt{3}) + (-8\sqrt{3} - 12) = 21 - 12 + 12\sqrt{3} - 8\sqrt{3}$ $= 9 + 4\sqrt{3}$

Final Answer:

$(g \circ f)(\sqrt{3}) = 9 + 4\sqrt{3}$

Would you like further clarification or details on any step?

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Here are 5 related questions:

1. What is the general process for evaluating composite functions?
2. How do you simplify expressions with square roots?
3. How does the distributive property apply to expanding terms like $(2\sqrt{3} + 3)^2$?
4. What is the domain of the composite function $g \circ f(x)$?
5. Can composite functions ever be non-existent for certain inputs?

Tip: When expanding binomials like $(a + b)^2$, remember to use the formula $a^2 + 2ab + b^2$ for faster calculations.