About This Quiz
Exponential Functions
Exponential functions are mathematical functions where the variable appears in the exponent. They are typically written in the form:
[latex] f(x) = a cdot b^x [/latex]
- a: The initial value of the function (the value when [latex] x = 0 [/latex]).
- b: The base of the exponential function. If [latex] b > 1 [/latex], the function represents exponential growth; if [latex] 0 < b < 1 [/latex], it represents exponential decay.
- Identify the Form of the Function: Recognize whether the problem involves an exponential growth or decay function. Growth functions have bases greater than 1, while decay functions have bases between 0 and 1.
- Understand Initial Values: The value of the function at [latex] x = 0 [/latex] is the initial value. This is often given directly or can be calculated by substituting [latex] x = 0 [/latex] into the function.
- Evaluate the Function: To find the value of the function at a specific point, substitute the given value of [latex] x [/latex] into the function. Use the properties of exponents to simplify the expression.
- Use Logarithms When Necessary: For more complex problems, such as finding the time required to reach a certain value, you may need to use logarithms to solve for the variable in the exponent. Remember the property [latex] log_b(a^c) = c cdot log_b(a) [/latex].
- Practice Multi-Step Problems: Some questions may involve multiple steps, such as evaluating a function at different points or combining multiple functions. Break down the problem into smaller parts and tackle each part systematically.