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Chapter 3: Problem 77

What is an exponential function?

Short Answer

Expert verified
An exponential function is a mathematical equation where a constant base is raised to a variable exponent. It has several unique properties and shows either exponential growth or decay depending on the base value.

Step by step solution

01

Define the Exponential Function

An exponential function is a specific type of function in mathematics, specifically in algebra, calculated by taking a number and raising it to the power of a variable. The general form is \(f(x) = a^x\), where \(a\) is a positive constant not equal to one, termed the 'base', and \(x\) is the exponent.
02

Discuss the Properties of Exponential Function

Exponential functions have several unique properties. Firstly, their rate of change (or derivative) is proportional to the function's current value, which leads to exponential growth or decay. Secondly, the function \(f(x) = a^x\) is always positive, regardless of the value of \(x\), assuming \(a\) is a positive number. Lastly, the base of the exponential function, \(a\), determines the behavior of the function. If \(a > 1\), the function is increasing, and if \(0 < a < 1\), the function is decreasing.
03

Provide Examples of Exponential Function

An example of an exponential function is \(f(x) = 2^x\). In this case, 2 is the base and \(x\) is the exponent. As \(x\) increases, the function will also increase. Another example could be \(f(x) = 0.5^x\), where the function will decrease as \(x\) increases.

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Most popular questions from this chapter

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

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.

Exercises \(153-155\) will help you prepare for the material covered in the next section. The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 a. Find Hungary's population, in millions, for 2006,2007 \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

Begin by graphing \(y=|x| .\) Then use this graph to obtain the graph of \(y=|x-2|+1 . \quad \text { (Section } 1.6, \text { Example } 3)\)

Consider the quadratic function $$ f(x)=-4 x^{2}-16 x+3 $$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.
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