The image you provided shows two exponential functions of the form $f(x) = b^x$, which illustrates the key features for different values of $b$. Let's summarize the two cases and generate the table, graphs, domain, range, and asymptotes for each case:

1. Case 1: $f(x) = b^x$, where $b > 1$

• This is an exponential growth function.

Table of Points:

Plot:

• For $b > 1$, you have three points:

  • $(-1, \frac{1}{b})$
  • $(0, 1)$ (the y-intercept)
  • $(1, b)$

• The curve increases rapidly as $x$ increases.

Domain and Range:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$
• Horizontal Asymptote: $y = 0$ (as $x \to -\infty$, the function approaches 0).

2. Case 2: $f(x) = b^x$, where $0 < b < 1$

• This is an exponential decay function.

Table of Points:

Plot:

• For $0 < b < 1$, the key points are:

  • $(-1, \frac{1}{b})$
  • $(0, 1)$
  • $(1, b)$

• The curve decreases rapidly as $x$ increases.

Domain and Range:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$
• Horizontal Asymptote: $y = 0$ (as $x \to \infty$, the function approaches 0).

Graph Summary:

• Left graph: Represents the exponential growth case $b > 1$.
• Right graph: Represents the exponential decay case $0 < b < 1$.

Would you like me to generate a plot for any specific values of $b$ or clarify any other details?

Related Questions:

1. How does changing the base $b$ affect the steepness of the exponential graph?
2. Can an exponential function ever cross the horizontal asymptote $y = 0$?
3. How can you determine whether an exponential function represents growth or decay just by looking at the base $b$?
4. What happens to the graph when $b = 1$?
5. What is the effect of negative exponents on the exponential function?

Tip:

Always remember, for exponential functions, the y-intercept is always $(0,1)$, regardless of the base $b$, since $b^0 = 1$ for any $b > 0$.