To determine which graph corresponds to which function, let's analyze the two functions and the graphs:

1. Function #1: $f(x) = \left(\frac{1}{3}\right)^x$
   This is an exponential decay function because the base, $\frac{1}{3}$, is less than 1. As $x$ increases, the function value decreases. As $x$ decreases, the function value increases. So, the graph should show a decreasing behavior as we move right along the $x$-axis and increasing as we move left.

2. Function #2: $f(x) = 3^x$
   This is an exponential growth function because the base, 3, is greater than 1. As $x$ increases, the function value grows rapidly, and as $x$ decreases, the function value approaches 0.

Graph Analysis:

• Graph i shows exponential growth: As $x$ increases, $y$ increases rapidly, suggesting this graph corresponds to an exponential growth function. Hence, Graph i represents Function #2: $f(x) = 3^x$.

• Graph ii shows exponential decay: As $x$ increases, $y$ decreases, which is consistent with the behavior of an exponential decay function. Therefore, Graph ii corresponds to Function #1: $f(x) = \left(\frac{1}{3}\right)^x$.

Conclusion:

• Graph i matches Function #2: $f(x) = 3^x$
• Graph ii matches Function #1: $f(x) = \left(\frac{1}{3}\right)^x$

Would you like more details or have any other questions?

Here are 5 related questions:

1. What is the general form of an exponential function?
2. How does changing the base of an exponential function affect its graph?
3. What happens to exponential growth functions as $x$ approaches infinity?
4. How can we distinguish between exponential growth and decay by looking at their graphs?
5. What are the real-world applications of exponential growth and decay?

Tip: Exponential functions with bases greater than 1 show growth, while those with bases between 0 and 1 show decay.