Problem 78 - College Algebra Essentials

Question

What is the natural exponential function?

Step-by-Step Solution

Answer

The natural exponential function, denoted as \( y = e^x \), is a function in mathematics where 'e' is a constant approximately equal to 2.71828. The function is always positive for any real value of 'x', increases without bound as 'x' increases, and approaches zero as 'x' approaches negative infinity.

Step 1: Definition

The natural exponential function is a function in mathematics that is defined as \( y = e^x \) where e is a mathematical constant approximately equal to 2.71828, and it is also known as Euler's number.

Step 2: Graph of the function

The graph of the natural exponential function y = e^x is always above the x-axis (y > 0 for all x). The function increases without bound as x increases (it tends towards positive infinity), and gets infinitely close to 0 as x approaches negative infinity.

Step 3: Growth rate and applications

The natural exponential function increases rapidly for positive x values. It is often used to model populations, investments, and in other areas of science and engineering where growth is proportional to the current amount.