*f*(*x*) = 

24 + 9*e*−*x* + 9*e*2*x* + 36*e*−3*x*

6 + 2*e*−*x* + 9*e*−3*x*

Given the function:

\[
f(x) = \frac{24 + 9e^{-x} + 9e^{2x} + 36e^{-3x}}{6 + 2e^{-x} + 9e^{-3x}}
\]

we aim to simplify or analyze this function.

### Step 1: Consider the Exponential Terms
Let's first explore the behavior of this function by factoring out the exponential terms where possible.

#### Numerator:
\[
24 + 9e^{-x} + 9e^{2x} + 36e^{-3x}
\]
We can try to factor the expression by grouping or by examining common exponential factors:
- The term \(24\) does not contain an exponential.
- \(9e^{-x}\) and \(9e^{2x}\) are multiplied by 9 and different powers of \(e\).
- \(36e^{-3x}\) is another distinct exponential term.

At first glance, these terms do not share a common exponential factor that would allow for straightforward simplification. So, let's leave the numerator in its current form.

#### Denominator:
\[
6 + 2e^{-x} + 9e^{-3x}
\]
Similarly, the terms in the denominator do not factor nicely either.

### Step 2: Analyze Special Cases

We can analyze the function \(f(x)\) by considering special cases or limits as \(x\) approaches specific values like \(x \to 0\), \(x \to \infty\), or \(x \to -\infty\).

#### As \(x \to 0\):
- \(e^0 = 1\), so substitute \(x = 0\) into the function:
\[
f(0) = \frac{24 + 9e^0 + 9e^0 + 36e^0}{6 + 2e^0 + 9e^0} = \frac{24 + 9 + 9 + 36}{6 + 2 + 9} = \frac{78}{17} \approx 4.588
\]

#### As \(x \to \infty\):
- As \(x\) increases, \(e^{-x} \to 0\) and \(e^{-3x} \to 0\), but \(e^{2x}\) grows rapidly.
- So the dominant term in the numerator is \(9e^{2x}\), and the remaining terms become negligible.
- The denominator tends toward 6 since \(2e^{-x}\) and \(9e^{-3x}\) tend to 0.
Thus, as \(x \to \infty\):
\[
f(x) \approx \frac{9e^{2x}}{6} = \frac{3}{2}e^{2x} \quad \text{(which grows exponentially)}
\]

#### As \(x \to -\infty\):
- As \(x\) decreases, \(e^{-x} \) and \(e^{-3x}\) dominate.
- The numerator becomes dominated by \(36e^{-3x}\) and \(9e^{-x}\).
- The denominator becomes dominated by \(9e^{-3x}\).

Thus, as \(x \to -\infty\):
\[
f(x) \approx \frac{36e^{-3x}}{9e^{-3x}} = 4
\]

### Summary
- For large positive \(x\), \(f(x)\) grows exponentially as \(e^{2x}\).
- For large negative \(x\), \(f(x)\) approaches a constant value of 4.
- At \(x = 0\), \(f(x)\) evaluates to approximately 4.588.

Would you like to explore any specific aspects of this function further, or is there another calculation you'd like to perform?

---

Do you want details or have any questions?  
Here are 5 related questions that expand on this topic:

1. What is the behavior of \(f(x)\) as \(x\) approaches negative infinity?
2. How does \(f(x)\) behave at intermediate values of \(x\), say \(x = 1\) or \(x = -1\)?
3. Can the function \(f(x)\) be rewritten in a more insightful form for specific ranges of \(x\)?
4. What are the roots of the equation \(f(x) = c\) for some constant \(c\)?
5. How would the function \(f(x)\) change if we adjusted the coefficients of the exponential terms?

**Tip:** When dealing with exponentials, understanding their growth rates (positive versus negative exponents) is crucial to analyzing the behavior of functions at extreme values of \(x\).

Explore the behavior of the function f(x) = (24 + 9e^{-x} + 9e^{2x} + 36e^{-3x}) / (6 + 2e^{-x} + 9e^{-3x}). Understand its asymptotic properties, special cases at x = 0, x approaching infinity, and x approaching negative infinity. Learn about the exponential growth and limits of f(x). Suitable for advanced high school students and anyone interested in exponential functions and limit analysis.

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