The task here is to match the given exponential functions with their corresponding graphs. The exponential functions provided are:

• a. $f(x) = (3^x)$
• b. $f(x) = \left( \frac{1}{3} \right)^x$
• c. $f(x) = 3^x$
• d. $f(x) = \left( \frac{1}{3} \right)^x$

General Analysis of Exponential Functions:

1. Exponential Growth (base greater than 1):

   • Functions like $f(x) = 3^x$ grow rapidly as $x$ increases. The graph passes through $(0, 1)$, and for large $x$, the function grows steeply upwards.
   • Graphs of the form $f(x) = 3^x$ or $f(x) = (3^x)$ will show this characteristic.

2. Exponential Decay (base between 0 and 1):

   • Functions like $f(x) = \left( \frac{1}{3} \right)^x$ show exponential decay, meaning as $x$ increases, the function approaches 0. The graph also passes through $(0, 1)$, but instead of increasing, it decreases.
   • Graphs of the form $f(x) = \left( \frac{1}{3} \right)^x$ or $f(x) = \left( \frac{1}{3} \right)^x$ will exhibit this behavior.

Matching the Graphs:

• First graph (top graph):

  • This graph shows exponential decay. It starts at $(0, 1)$ and approaches 0 as $x$ increases.
  • This corresponds to the formula $f(x) = \left( \frac{1}{3} \right)^x$ (Option b or d).

• Second graph:

  • This graph shows exponential growth. As $x$ increases, the function increases rapidly and grows steeper.
  • This corresponds to the formula $f(x) = 3^x$ (Option c or a).

• Third graph:

  • This graph also shows exponential decay, similar to the first graph. It starts at $(0, 1)$ and decreases as $x$ increases.
  • This corresponds to another $\left( \frac{1}{3} \right)^x$ function (Option b or d).

• Fourth graph (bottom graph):

  • This graph shows exponential growth, similar to the second graph, increasing rapidly for positive $x$.
  • This corresponds to another $3^x$ function (Option c or a).

Final Matching:

• First graph: b. $f(x) = \left( \frac{1}{3} \right)^x$
• Second graph: c. $f(x) = 3^x$
• Third graph: d. $f(x) = \left( \frac{1}{3} \right)^x$
• Fourth graph: a. $f(x) = (3^x)$

Would you like further explanation or details on these matches?

Here are 5 related questions you might explore:

1. What is the difference between exponential growth and decay?
2. How does changing the base of an exponential function affect its graph?
3. Can exponential functions have horizontal asymptotes?
4. What happens to the graph of $3^x$ when a constant is added or subtracted from $x$?
5. How do exponential functions compare with logarithmic functions?

Tip: For exponential functions, if the base is greater than 1, the function grows; if the base is between 0 and 1, the function decays.