Problem 77 - College Algebra Essentials

Question

What is an exponential function?

Step-by-Step Solution

Answer

An exponential function is a mathematical function in which the input (or variable) is located in the exponent. The general form of an exponential function is \( f(x) = a \cdot b^{x} \) where 'a' is any constant, 'b' is a positive real number different from 1, and 'x' is the variable located in the exponent. The base, 'b', indicates whether the function will depict exponential growth ('b' is greater than 1) or decay ('b' is less than 1).

Step 1: Introducing The Concept

An exponential function is a mathematical function, wherein the input (x) of the function appears in the exponent. This is a relation between two numbers, also known as variables, so that one quantity depends on another.

Step 2: Explaining The General Form

The general form of an exponential function is \( f(x) = a \cdot b^{x} \) where 'a' is a constant, 'b' is a positive real number, and 'b' is not equal to 1. In this function, 'b' is called the base of the exponential function.

Step 3: Describing The Characteristics

Exponential functions have several important characteristics. For instance, the base of an exponential function is always positive and different from 1. Moreover, the graph of an exponential function always shows the exponential growth or exponential decay depending on whether the base is greater than 1 or less than 1, respectively.