Section 4.1: Exponential Functions

1. Evaluate an exponential function \(f(x) = a^{x}\) at integer and rational values of \(x\).
2. Specify the domain and range of an exponential function.
3. Determine whether a given function is an exponential function.
4. Sketch the graph of an exponential function \(f(x) = a^{x}\), including its behavior as \(x\) approaches both positive and negative infinity.
5. Given the graph of a function \(f(x)\), sketch the graphs of \(g(x) = f(x) + c\), \(g(x) = f(x - c)\), \(g(x) = -f(x)\), and \(g(x) = f(-x)\).
6. Use (but do not memorize!) the formula for compound interest to find the amount of money in a savings account if the account starts with a principal \(P\), has yearly interest rate \(r\), and is compounded \(n\) times per year for a total of \(t\) years.

Section 4.2: The Natural Exponential Function

1. Give the decimal expansion of the number \(e\) to 3 decimal places.
2. Sketch the graph of an exponential function \(f(x) = e^{x}\), including its behavior as \(x\) approaches both positive and negative infinity.