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Chapter 4: Problem 77

What is an exponential function?

Short Answer

Expert verified
An exponential function is a mathematical function of the form \(f(x) = b^{x}\), where \(b\) is a positive real number not equal to 1.

Step by step solution

01

Definition of Exponential Function

An exponential function is a mathematical function of the form \(f(x) = b^{x}\), where \(b\) is a positive real number, and \(b \neq 1\). \(b\) is referred to as the 'base' and \(x\) is the exponent.
02

Properties of Exponential Functions

1. The base \(b\) is always positive and not equal to 1. \n2. The function is increasing if \(b > 1\) and decreasing if \(01\) and \( \lim_{x \to \infty} b^{x} = 0 \) if \(0<b<1\).\n5. The y-intercept of the function on the graph is always (0, 1) regardless of the base.
03

Visualizing Exponential Functions

In a graphical representation, an exponential function is a curved line that gets nearer to the x-axis but never touches it as \(x\) decreases, and rises rapidly as \(x\) increases.\nThe curve passes through the point (0,1) representing the y-intercept.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ \log (x+3)+\log x=1 $$

Explaining the Concepts Suppose that a population that is growing exponentially increases from \(800,000\) people in 2010 to \(1,000,000\) people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ 2^{x+1}=8 $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln \sqrt{2}=\frac{\ln 2}{2} $$
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