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The given function is:

\[
f(x) = \frac{3 \cdot 5^x + 7 \cdot 8^{-x}}{8 \cdot 5^x + 7 \cdot 8^{-x}}
\]

### Step 1: Simplify the function
Let’s set some substitutions for simplification. Let:
\[
y = 5^x
\quad \text{and} \quad z = 8^{-x} = \frac{1}{8^x}.
\]

So, the function becomes:

\[
f(x) = \frac{3y + 7z}{8y + 7z}
\]

### Step 2: Analyze the behavior
- As \( x \to \infty \), \( 5^x \to \infty \) and \( 8^{-x} \to 0 \), so the function approximates:
  \[
  f(x) \to \frac{3y}{8y} = \frac{3}{8}.
  \]

- As \( x \to -\infty \), \( 5^x \to 0 \) and \( 8^{-x} \to \infty \), so the function approximates:
  \[
  f(x) \to \frac{7z}{7z} = 1.
  \]

### Step 3: Check for critical points
We check if the function takes other values between \( \frac{3}{8} \) and \( 1 \). The function is continuous for all \( x \), and the numerator and denominator are positive for all real \( x \), meaning \( f(x) \) is strictly between \( \frac{3}{8} \) and \( 1 \).

### Step 4: Conclusion
The range of the function is the interval between the two limiting values:

\[
\boxed{\left(\frac{3}{8}, 1\right)}
\]

Would you like further clarification, or have any specific questions?

Here are five follow-up questions you could explore:
1. What happens if we change the base of the exponents in the function?
2. How would you find the inverse of this function if it exists?
3. Can you derive a similar range for a similar function with different constants?
4. What does the graph of this function look like across the domain of all real numbers?
5. How would the function change if you had complex exponents instead?

**Tip**: When dealing with rational functions with exponential terms, always consider the behavior at extreme values of \(x\) to predict the function's range and limits.

Find the range of the function f(x) = (3 * 5^x + 7 * 8^{-x}) / (8 * 5^x + 7 * 8^{-x}).

This problem involves finding the range of a rational function with exponential terms. It includes the analysis of limits as x approaches positive and negative infinity, helping to determine that the range is between 3/8 and 1. This content is suitable for high school students studying Pre-Calculus or Calculus.

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