Given the function:

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

Our goal is to simplify or analyze this function. Let's start by examining both the numerator and the denominator.

Step 1: Factor the Numerator and Denominator

Numerator:

$24 + 9e^{-x} + 9e^{2x} + 36e^{-3x}$ This can be factored by grouping terms and finding common factors:

= 9e^{-x}(e^{2x} + e^{-x}) + 24 + 36e^{-3x}$$ But this expression doesn't seem to factor further in a straightforward way, so we'll leave it as is for now. #### Denominator: $$6 + 2e^{-x} + 9e^{-3x}$$ This expression does not factor in a simple way either, so we'll also leave it as it is. ### Step 2: Simplification or Further Analysis Since the terms in the numerator and denominator don't simplify easily through factoring, we can instead evaluate the function for specific values of $$x$$ or explore its behavior (such as limits at $$x \to \infty$$ or $$x \to -\infty$$) if needed. Would you like to proceed with evaluating this function at specific points, analyzing its limits, or exploring any other aspects? --- Do you want details or have any questions? Here are 5 related questions that expand on this topic: 1. What is the behavior of $$f(x)$$ as $$x$$ approaches infinity? 2. How can we find the roots of the function $$f(x) = 0$$? 3. What are the first and second derivatives of $$f(x)$$ with respect to $$x$$? 4. Can the numerator and denominator be simplified further by considering a common exponential factor? 5. How does the function $$f(x)$$ behave as $$x$$ approaches negative infinity? **Tip:** When dealing with complex functions involving exponentials, examining the behavior at extreme values (like $$x \to \infty$$ and $$x \to -\infty$$) can provide insights into the function's long-term behavior.