Let's break the problem into parts and solve step by step. The problem involves filling out tables and analyzing the exponential function $f(x) = 3^x$ and the logarithmic function $g(x) = \log_3 x$.

Step 1: Fill in the table for $f(x) = 3^x$

Using the exponential formula $f(x) = 3^x$, calculate $f(x)$ for each value of $x$:

• $x = -2$: $f(x) = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
• $x = -1$: $f(x) = 3^{-1} = \frac{1}{3}$
• $x = 0$: $f(x) = 3^0 = 1$
• $x = 1$: $f(x) = 3^1 = 3$
• $x = 2$: $f(x) = 3^2 = 9$

So, the $f(x)$ table is:

Step 2: Fill in the table for $g(x) = \log_3 x$

Using the logarithmic formula $g(x) = \log_3 x$, calculate $g(x)$ for each value of $x$. Recall that $\log_3 x$ is the power to which $3$ must be raised to get $x$:

• $x = \frac{1}{9}$: $g(x) = \log_3 \frac{1}{9} = -2$
• $x = \frac{1}{3}$: $g(x) = \log_3 \frac{1}{3} = -1$
• $x = 1$: $g(x) = \log_3 1 = 0$
• $x = 3$: $g(x) = \log_3 3 = 1$
• $x = 9$: $g(x) = \log_3 9 = 2$

So, the $g(x)$ table is:

Step 3: Fill in attributes of $f(x)$ and $g(x)$

For $f(x) = 3^x$:

1. Domain: $(-\infty, \infty)$ (all real numbers)
2. Range: $(0, \infty)$ (positive real numbers)
3. x-intercept: None (never crosses the x-axis)
4. y-intercept: $(0, 1)$
5. Asymptote: Horizontal asymptote at $y = 0$

For $g(x) = \log_3 x$:

1. Domain: $(0, \infty)$ (positive real numbers)
2. Range: $(-\infty, \infty)$ (all real numbers)
3. x-intercept: $(1, 0)$
4. y-intercept: None (never crosses the y-axis)
5. Asymptote: Vertical asymptote at $x = 0$

Step 4: Plot the Graph

The graph includes:

1. The exponential function $f(x) = 3^x$
2. The logarithmic function $g(x) = \log_3 x$
3. The line $y = x$ (as a reference line showing the inverse relationship).

If you'd like, I can draw the graph and display it for better understanding.

Would you like the graph or need more clarifications?

5 Related Questions:

1. What is the relationship between $f(x)$ and $g(x)$?
2. Why does $g(x) = \log_3 x$ have a vertical asymptote at $x = 0$?
3. Can $g(x) = \log_3 x$ be defined for negative $x$?
4. How does the base of the exponential function affect the graph of $g(x)$?
5. What happens to the graphs if the base changes from 3 to 2?

Tip: Always remember that exponential and logarithmic functions are inverses of each other!