Evaluate the composite function, where

*f*(*x*) = 2*x* + 3, and *g*(*x*) = *x*2 − 4*x*.

(*g* ∘ *f*)(

sqrt 3

)

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.

Evaluate the composite function, where f(x) = 2x + 3, and g(x) = x^2 − 4x. (g ∘ f)(sqrt(3))

Learn how to evaluate the composite function (g ∘ f)(sqrt(3)) step-by-step. This problem covers key algebra concepts such as function composition and expanding binomials, including the use of square roots. Suitable for high school students in Grades 10-12.

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