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Chapter 3: Problem 77

What is an exponential function?

Short Answer

Expert verified
An exponential function is a mathematical function of the form \(f(x) = a \cdot b^{x}\), where \(a\) and \(b\) are constants, \(b > 0\), \(b \neq 1\), and \(x\) is a real number. The function rapidly increases or decreases in value depending on the base \(b\), and has a y-intercept of \(a\).

Step by step solution

01

Definition

An exponential function is a function of the form \(f(x) = a \cdot b^{x}\), where \(a\) and \(b\) are constants, \(b > 0\), \(b \neq 1\), and \(x\) any real number.
02

Characteristic Properties of an Exponential Function

Exponential functions have several important properties which characterizes them: \n1. The graph of an exponential function is always upwards or downwards sloping depending on whether \(b > 1\) or \(0 < b < 1\). \n2. The \(y\)-intercept of the graph is \(a\), where \(a\) is the coefficient of \(b^x\). \n3. The rate of increase or decrease is proportional to its current value. The larger the value of \(b\), the steeper the curve.
03

Examples of Exponential Functions

Here are some examples of exponential functions: \n1. \(f(x) = 2^{x}\) - This is an exponential function with base 2. The graph of this function is always increasing. \n2. \(f(x) = 3(0.5)^x\) - This is an exponential function with base 0.5 and a factor of 3. The graph of this function is always decreasing.

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Will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)

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