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Chapter 3: Problem 103

In Exercises 101 - 104, sketch the graphs of \( f \) and \( g \) and describe the relationship between the graphs of \( f \) and \( g \). What is the relationship between the functions \( f \) and \( g \)? \( f(x) = e^x \), \( g(x) = \ln x \)

Short Answer

Expert verified
The function \(g(x) = \ln x\) is the inverse of the function \(f(x) = e^x \). Their graphs are mirror images with respect to the line \(y = x\).

Step by step solution

01

Sketching the Graphs

Begin by plotting the graphs of both functions. For \( f(x) = e^x \), the function increases rapidly for positive x, crosses the y-axis at \( y = 1 \) and approaches 0 as \( x \) becomes more and more negative. For \( g(x) = \ln x \), the function crosses the x-axis at \( x = 1 \) and increases slowly as \( x \) becomes larger. The graph is undefined for \( x \leq 0 \).
02

Observing the Relationship Between the Graphs

The graph of \( f = e^x \) is the mirror image of the graph of \( g = \ln x \) in the line \(y = x\). This is a reflection property between the logarithmic function and the exponential function.
03

Determining the Relationship Between the Functions

Function \( g \) is the inverse of function \( f \). This is demonstrated mathematically by noting that \( e^{\ln x} = x \) for all \( x > 0 \) and \( \ln(e^x) = x \) for all \( x \) in real numbers. This indicates that indeed \( f \) is the inverse function of \( g \) and vice versa.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function
The exponential function, often written as \( f(x) = e^x \), is a fundamental mathematical function where the base \( e \) is an irrational constant approximately equal to 2.71828. This function is essential in various fields such as biology, economics, and physics due to its unique property of constant growth rate. Here鈥檚 what you need to know about the behavior of an exponential function:
  • For positive values of \( x \), the function increases rapidly. This is due to the fact that each increment in \( x \) results in an ever-increasing output, a hallmark of exponential growth.

  • At \( x = 0 \), the value of \( e^x \) is 1, meaning the graph crosses the y-axis at 1.

  • As \( x \) becomes more negative, \( e^x \) approaches zero but never actually reaches it, demonstrating an asymptotic behavior towards the x-axis.
This rapid increase and asymptotic behavior make the exponential function unique compared to linear or quadratic functions.
Logarithmic Function
The logarithmic function, denoted as \( g(x) = \ln x \), is the inverse of the exponential function. Understanding logarithms is crucial as they simplify multiplication and powers into addition and subtraction, making complex calculations more manageable. Here鈥檚 how a basic logarithmic function behaves:
  • The function is undefined for \( x \leq 0 \), meaning it only takes positive real numbers.

  • It crosses the x-axis at \( x = 1 \), which aligns with the fact that the logarithm of 1 is 0 for any base.

  • As \( x \) increases, \( \ln x \) grows, but at a decreasing rate, indicating that it increases very slowly.
The logarithmic function is crucial because its reflective property in relation to the exponential function adds depth to understanding how these functions interact as inverses. It鈥檚 this reflection that is observed across the line \( y = x \), signifying the inverse relationship.
Graph Sketching
Graph sketching is an essential skill in mathematics that involves plotting functions to visualize their structures and behaviors. When sketching graphs for \( f(x) = e^x \) and \( g(x) = \ln x \), there are a few key points to remember:
  • Start by identifying crucial points such as intersects and asymptotes. For \( e^x \), note that it intersects the y-axis at 1. For \( \ln x \), it intersects the x-axis at 1.

  • Understand the general shape. The graph of \( e^x \) rises sharply, whereas \( \ln x \) rises slowly and levels off.

  • Pay attention to the reflection property. Because these functions are inverses, their graphs reflect across the line \( y = x \). This can help in verifying the accuracy of sketched graphs.
Being able to translate the mathematical expressions of functions into graphical representations enhances comprehension of their properties and relationships. It also provides a visual confirmation of theoretical concepts such as the inverse relationship between exponential and logarithmic functions.

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Most popular questions from this chapter

In Exercises 139 - 142, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the sum of two numbers is equal tothe product of the logarithms of the numbers.

In Exercises 69 - 74, use the acidity model given by \( pH = -\log \left[H^+\right] \), where acidity \( (pH) \) is a measure of the hydrogen ion concentration \( \left[H^+\right] \) (measured in moles of hydrogen per liter) of a solution. find the \( pH \) if \( \left[H^+\right] = 2.3 \times 10^{-5} \).

At \( 8:30 \) A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the persons temperature twice. At \( 9:00 \) A.M. the temperature was \( 85.7^\circ F \) and at \( 11:00 \) A.M. the temperature was \( 82.8^\circ F \). From these two temperatures,the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula \( t = -10 ln \dfrac{T - 70}{98.6 - 70} where \) t \( is the time in hours elapsed since the person died and \) T \( is the temperature (in degrees Fahrenheit) of the persons body. (This formula is derived from a general cooling principle called Newtons Law of Cooling. It uses the assumptions that the person had a normal body temperature of \) 98.6^\circ F \( at death, and that the room temperature was a constant \) 70^\circ F $. ) Use the formula to estimate the time of death of the person.

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