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: $g(x) \approx 6.363 \cdot (0.707)^x$

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.