Problem 58 - College Algebra

Question

What is an exponential function?

Step-by-Step Solution

Answer

An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \) where 'a' and 'b' are constants, 'b' is greater than zero and not equal to one, and 'x' is a variable.

Step 1: Define Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \), where 'a' and 'b' are real numbers, and \( b > 0, b \neq 1 \), and 'x' is any real number. The base, 'b', is constant and the exponent, 'x', is a variable.

Step 2: Explain Characteristics of Exponential Function

Exponential functions are characterized by their rapid growth or decay, depending on the value of the base 'b'. If \( b > 1 \), the function 'f(x)' represents exponential growth, while if \( 0 < b < 1 \), it represents exponential decay.

Step 3: Provide an Example of Exponential Function

A simple example of an exponential function is \( f(x) = 2^x \). Here, the base is '2', which is a constant and the exponent 'x' is the variable. As 'x' increases, the function f(x) grows exponentially.