This image contains several mathematical questions related to exponential functions, graph analysis, and properties of exponential equations.

Let's break down the questions:

1. Question 30: Which function increases the fastest? This question gives you four functions:

   • A. $y = 104x$
   • B. $y = -2 \cdot 15^x$
   • C. $y = 12^x$
   • D. $y = -220x$

   To determine which function increases the fastest, you'd look at the rate of growth for each. Exponential functions, like $y = 12^x$ and $y = -2 \cdot 15^x$, typically grow faster than linear functions such as $y = 104x$ and $y = -220x$. Among the exponential functions, $y = -2 \cdot 15^x$ will increase the fastest because the base of 15 is greater than 12.

2. Question 31: Examine the graphs of $f(x) = 3^x$ and $g(x) = 5x$. This question asks you to estimate where one function is greater than the other:

   • (a) Estimate the values of $x$ where $f(x) > g(x)$.

   • (b) Estimate the values of $x$ where $g(x) > f(x)$.

   • (c) As $x$ decreases, the graph of $f(x)$ approaches 0. Will it ever intersect the x-axis?

   • (a) $f(x) = 3^x$ is an exponential function, so it grows rapidly as $x$ increases. $g(x) = 5x$ is linear. You will estimate when the graph of $3^x$ is above $5x$.

   • (b) You will estimate when the linear graph of $g(x) = 5x$ is above $f(x) = 3^x$.

   • (c) As $x$ decreases for $f(x) = 3^x$, the graph approaches 0 but never touches the x-axis, meaning $f(x)$ will not intersect the x-axis.

3. Question 32: Why can't the value of $a$ in an exponential function be 0? Why can't $b$ be equal to 1?

   • The base $b$ of an exponential function must be greater than 0 and not equal to 1. If $b = 1$, the function would always be constant and not exhibit exponential growth or decay.
   • The value of $a$ (the coefficient) can be any real number, but the base $b$ must satisfy the conditions mentioned above.

Would you like help with a deeper explanation on any of these points or solving other related problems?

Further Questions:

1. How do exponential functions behave in terms of growth for various values of $b$?
2. What are the properties of linear vs exponential growth?
3. Can exponential functions ever have negative values? Why or why not?
4. How do transformations affect the graph of an exponential function?
5. How do you graph exponential functions without using a calculator?

Tip: When analyzing exponential functions, remember that they either grow or decay depending on whether the base is greater than 1 or between 0 and 1.