Let's analyze the two functions given in the image to understand their behavior and sketch them:

a. $f(x) = e^x$

• This is the exponential growth function.
• It has the following key characteristics:
  • Domain: $(-\infty, \infty)$ (the function is defined for all real $x$).
  • Range: $(0, \infty)$ (the output is always positive).
  • It passes through the point $(0,1)$, because $e^0 = 1$.
  • As $x \to \infty$, $e^x \to \infty$ (exponential growth).
  • As $x \to -\infty$, $e^x \to 0$ (but never reaches zero).
  • The graph is increasing and has a horizontal asymptote at $y = 0$.

b. $f(x) = e^{-x}$

• This is the exponential decay function.
• It has the following key characteristics:
  • Domain: $(-\infty, \infty)$ (defined for all real $x$).
  • Range: $(0, \infty)$ (always positive).
  • It passes through the point $(0, 1)$ because $e^{-0} = 1$.
  • As $x \to \infty$, $e^{-x} \to 0$ (exponentially decays to 0 but never reaches it).
  • As $x \to -\infty$, $e^{-x} \to \infty$ (exponential growth for negative values of $x$).
  • The graph is decreasing and also has a horizontal asymptote at $y = 0$.

Would you like me to create an actual sketch of these functions for you?

Here are 5 related questions to explore further:

1. What are the differences between exponential growth and decay in real-world applications?
2. How do we derive the derivative of $e^x$ and $e^{-x}$?
3. How does the horizontal asymptote affect the behavior of these exponential functions?
4. What happens to these functions when we apply transformations such as shifting or scaling?
5. Can we solve for $x$ if $e^x = a$ for some constant $a$?

Tip: When analyzing exponential functions, always check the base of the exponent. For $e^x$, $e \approx 2.718$, which drives the growth or decay of the function.