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To find the exponential functions that pass through each pair of points, we assume the form:

\[
f(x) = ab^x
\]

### Problem (a)

For \( f(x) \), we are given the points \((-1, \frac{4}{3})\) and \( (4, 324) \).

1. Substitute \((-1, \frac{4}{3})\):
   \[
   \frac{4}{3} = ab^{-1} = \frac{a}{b}
   \]
   which gives us: \( a = \frac{4}{3} b \).

2. Substitute \( (4, 324) \):
   \[
   324 = ab^4
   \]
   Substitute \( a = \frac{4}{3} b \) into this equation:
   \[
   324 = \frac{4}{3} b \cdot b^4 = \frac{4}{3} b^5
   \]
   \[
   b^5 = \frac{324 \cdot 3}{4} = 243
   \]
   Taking the fifth root of both sides:
   \[
   b = 3
   \]
3. Substitute \( b = 3 \) back to find \( a \):
   \[
   a = \frac{4}{3} \cdot 3 = 4
   \]

So, the function \( f(x) \) is:
\[
f(x) = 4 \cdot 3^x
\]

### Problem (b)

For \( g(x) \), we are given the points \((-1, 9)\) and \( (3, 2) \).

1. Substitute \((-1, 9)\):
   \[
   9 = \frac{a}{b}
   \]
   which implies \( a = 9b \).

2. Substitute \( (3, 2) \):
   \[
   2 = ab^3
   \]
   Substitute \( a = 9b \) into this equation:
   \[
   2 = 9b \cdot b^3 = 9b^4
   \]
   \[
   b^4 = \frac{2}{9}
   \]
   Taking the fourth root of both sides:
   \[
   b = \sqrt[4]{\frac{2}{9}} \approx 0.707
   \]
3. Substitute \( b \approx 0.707 \) back to find \( a \):
   \[
   a = 9 \cdot 0.707 \approx 6.363
   \]

So, the function \( g(x) \) is approximately:
\[
g(x) \approx 6.363 \cdot (0.707)^x
\]

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Would you like additional details or have questions about this solution?

### Related Questions:
1. How do you find the exponential function if you only have one point and a known asymptote?
2. How does changing the base \( b \) affect the behavior of an exponential function?
3. What is the difference between exponential growth and decay?
4. How can you identify an exponential function from a graph?
5. How do logarithmic functions relate to exponential functions?

### Tip:
When solving exponential equations, try expressing both sides with a common base if possible; it simplifies solving powers and roots.

For each of the following, find the formula for an exponential function that passes through the two points given.

a. (-1, 4/3) and (4, 324)
b. (-1, 9) and (3, 2)


Learn how to find the formula for an exponential function that passes through two points, using points such as (-1, 4/3) and (4, 324) or (-1, 9) and (3, 2). This example involves understanding exponential functions, solving systems, and using roots and powers.

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