To solve the given problem, we need to find the exponential functions of the form:

$f(x) = ab^x$

where $a$ is the initial value (when $x = 0$), and $b$ is the base or growth/decay factor.

Part (a): Points (0, 7) and (4, 16807)

1. When $x = 0$, $f(x) = ab^0 = a$. From the point $(0, 7)$, we know that $a = 7$.
2. Now, using the point $(4, 16807)$, substitute into the equation: $16807 = 7b^4$
3. Solve for $b$: $b^4 = \frac{16807}{7} = 2401$ $b = \sqrt[4]{2401} = 7$

Thus, the exponential function for part (a) is:

$f(x) = 7 \cdot 7^x$

Part (b): Points (0, 14256) and (4, 11)

1. Again, for $x = 0$, $g(x) = ab^0 = a$. From the point $(0, 14256)$, we know that $a = 14256$.
2. Using the point $(4, 11)$, substitute into the equation: $11 = 14256b^4$
3. Solve for $b$: $b^4 = \frac{11}{14256} = \frac{1}{1296}$ $b = \sqrt[4]{\frac{1}{1296}} = \frac{1}{6}$

Thus, the exponential function for part (b) is:

$g(x) = 14256 \cdot \left(\frac{1}{6}\right)^x$

Would you like more details on how to solve exponential functions, or do you have any questions? Here are five related questions to explore further:

1. How do you determine if an exponential function represents growth or decay?
2. Can you find the inverse of an exponential function like $f(x) = 7 \cdot 7^x$?
3. How does the base $b$ affect the shape of the exponential graph?
4. What are the key differences between exponential growth and logarithmic growth?
5. How do you transform an exponential function to shift it horizontally or vertically?

Tip: When working with exponential equations, always check if the base is a fraction or a number greater than 1 to identify whether the function represents growth or decay.