Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

What is an exponential function?

Step-by-Step Solution

Answer

An exponential function is a mathematical function which involves a fixed base raised to the power of a variable exponent. It is widely applied in real-life situations such as population growth or interest computation, and its graph has characteristic properties.

Step 1: Definition

An exponential function is considered a mathematical function where a fixed base, greater than 1, is raised to the power of a variable exponent.

Step 2: Mathematical Representation

The standard form of an exponential function is \( f(x) = a \cdot b^{x} \) where \( a \) can be any real number and \( b \) is a positive real number other than 1, known as the base.

Step 3: Real-life applications

Exponential functions are used to model a variety of real-life situations such as population growth, radioactive decay, interest computation in finance, to name a few.

Step 4: Key properties

1. The graph of an exponential function always has a horizontal asymptote (a line that a curve approaches, as it heads towards infinity), which is the x-axis, i.e., \( y = 0 \).\n 2. The function is always above the x-axis (positive) but can never touch or cross it.\n 3. The function is always increasing (for \( b > 1 \)) or decreasing (for \( 0 < b < 1 \)).