What happens when “x” is a negative value? Show this with a sketch f(x) = 2^x and f(x) = 2^-x 

The plot above shows the graphs of \(f(x) = 2^x\) (in blue) and \(f(x) = 2^{-x}\) (in red, dashed). Here's a summary of their behavior:

- **For \(f(x) = 2^x\)**: As \(x\) becomes negative, the values get smaller and approach zero but never reach it. As \(x\) increases positively, the function grows rapidly.
- **For \(f(x) = 2^{-x}\)**: This function is a reflection of \(2^x\) over the y-axis. It decreases rapidly as \(x\) increases, but approaches zero for large negative values of \(x\).

Would you like more details, or have any questions about these functions?

Here are 5 related questions you might find helpful:
1. How does the behavior of \(3^x\) compare to \(2^x\)?
2. What happens to the function \(2^x\) when \(x\) approaches infinity?
3. What is the derivative of \(2^x\) and how does it relate to the function's growth?
4. Can you describe the behavior of \(f(x) = a^x\) for values of \(a\) between 0 and 1?
5. How do the graphs of logarithmic functions relate to these exponential functions?

**Tip**: When \(x\) is negative in an exponential function \(a^x\), the result is equivalent to \( \frac{1}{a^{|x|}} \), which tends to 0 as \(x\) becomes more negative for \(a > 1\).

What happens when 'x' is a negative value? Show this with a sketch f(x) = 2^x and f(x) = 2^-x.

Explore how exponential functions behave when 'x' is negative, focusing on f(x) = 2^x and its reflection f(x) = 2^-x. This analysis includes key concepts such as exponential growth, decay, and reflection across the y-axis. Ideal for students in Grades 9-12.

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