Introduction to Exponential Functions: Videos & Practice Problems

An exponential function is a mathematical expression where a constant base is raised to a variable exponent, such as $2^x$. This contrasts with polynomial functions, where the variable is the base and raised to a constant exponent, like $x^2$. The key difference lies in the placement of the variable: in exponential functions, the variable is in the exponent, while in polynomial functions, it is the base. Exponential functions grow or decay at rates proportional to their current value, leading to rapid increases or decreases, unlike the more predictable growth of polynomial functions.

To determine if a function is exponential, check the base and the exponent. The base must be a constant, positive number, and not equal to 1. The exponent should contain a variable, indicating that it can change. For example, in the function ${\frac{2}{3}}^x$, the base $\frac{2}{3}$ is constant and positive, and the exponent $x$ is a variable, making it an exponential function. Conversely, a function like $1^y$ is not exponential because the base is 1.

To evaluate an exponential function for a given value of $x$, substitute the value into the function and compute the result. For example, if the function is $2^x$ and $x$ is 4, calculate $2^4$, which equals 16. For negative or decimal exponents, use the rules of exponents or a calculator. For instance, $2^{-3}$ is $\frac{1}{8}$, and $2^{3.14}$ can be calculated using a calculator, yielding approximately 8.815.

The base of an exponential function must meet specific criteria: it must be a constant, positive number, and it cannot be equal to 1. These characteristics ensure that the function exhibits exponential growth or decay. A base of 1 would result in a constant function, as any number raised to any power of 1 remains 1. For example, in the function ${10}^x$, the base 10 is constant, positive, and not equal to 1, making it suitable for an exponential function. These properties are crucial for the function to maintain its exponential nature.

To evaluate exponential functions with large or decimal exponents using a calculator, first enter the base. Then, use the caret key (^) to indicate raising to a power. Enter the exponent, ensuring accuracy by using parentheses if needed. For example, to calculate $2^{12}$, type 2, then the caret key, followed by 12, resulting in 4096. For decimal exponents like $2^{3.14}$, follow the same steps, yielding approximately 8.815. This method ensures precise calculations, especially for complex or non-integer exponents.