Supply a valid technology formula for the given function.
3
1 − 2−8x

To express the given function \( f(x) = \frac{3}{1 - 2^{-8x}} \) in a valid technology formula (for graphing calculators or mathematical software), we need to ensure it's entered correctly. In general, technology formulas for calculators follow a specific syntax depending on the tool being used.

For this function:

- The numerator is \( 3 \),
- The denominator is \( 1 - 2^{-8x} \).

The valid technology formula (assuming you're using a tool that supports exponentials, like Desmos or a graphing calculator) would be written as:

\[
f(x) = \frac{3}{1 - 2^{(-8x)}}
\]

Here’s the breakdown:
- The exponent \( -8x \) is placed within parentheses to ensure the proper order of operations.
- The entire denominator is in parentheses to group \( 1 - 2^{-8x} \) as a single unit.

### Would you like to go deeper into how this function behaves, such as discussing asymptotes or other properties?

Here are 5 related questions:
1. How does the function behave as \( x \) approaches positive or negative infinity?
2. What are the asymptotes of this function?
3. Can this function have any real zeros (where \( f(x) = 0 \))?
4. How would the graph of this function look, and what key points should we analyze?
5. How does changing the exponent \( -8x \) affect the shape of the function?

**Tip:** Always use parentheses in technology formulas to ensure correct order of operations, especially when dealing with exponents or fractions.

Supply a valid technology formula for the given function. 3 / (1 − 2^-8x)

Understand how to express the function 3 / (1 − 2^-8x) as a valid technology formula for graphing calculators. This problem covers exponential functions and algebraic fractions, suitable for students in Grades 10-12.

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