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?

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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 $x$.