/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 95 For each function given below, (... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 95

For each function given below, (a) determine the domain and the range, (b) set an appropriate window, and (c) draw the graph. Graphs may vary, depending on the scale used. $$ f(x)=3.4 \ln x-0.25 e^{x} $$

Short Answer

Expert verified
Domain: \( x > 0 \). Range: \( f(x) \in (-\infty, \infty) \). Suggested window: \( x \) from 0.1 to 5, \( y \) from -5 to 5.

Step by step solution

01

Determine the domain

The function is given as \(f(x) = 3.4 \ln x - 0.25 e^x\). The natural logarithm function \( \ln x \) is defined only for positive values of \( x \). Therefore, the domain of the function is all positive real numbers, \( x > 0 \).
02

Determine the range

To determine the range, plot the function or analyze its behavior. The exponential term \( e^x \) grows very fast and dominates \( 3.4 \ln x \) for large values of \( x \), driving the function to negative infinity. For very small positive values of \( x \), \( \ln x \) becomes very negative, but the exponential term remains small. Thus, the range of the function is all real numbers, \( f(x) \in (-\infty, \infty) \).
03

Set an appropriate window

Choose a window where the significant features of the function are visible. A good starting window might be \( x \) from 0.1 to 5 to avoid the undefined region at \( x=0 \) and to observe the behavior as \( x \) increases. For the \( y \)-axis, start with \( y \) from -5 to 5 to capture sufficient range and adjust as necessary based on the behavior of the function.
04

Draw the graph

Plot the function using graphing software or a graphing calculator with the chosen window. Plot several points within the window to ensure the shape of the function is captured accurately. Verify the behavior as \( x \) approaches the boundaries of the window.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Domain of a Function
When we talk about the domain of a function, we're essentially asking the question: 'For which values of x is our function valid?' In the given function, which is expressed as \( f(x) = 3.4 \ln x - 0.25 e^x \), we need to consider the restrictions of both the natural logarithm and the exponential functions.
The natural logarithm function, \( \ln x \), is defined only for positive values of \( x \). Therefore, our function \( f(x) \) is valid only when \( x \) is greater than zero, specified mathematically as \( x > 0 \). Hence, the domain of this function is all positive real numbers. Imagine a number line where we start just above zero and go all the way to positive infinity.
Range of a Function
The range of a function concerns itself with the possible output values (y-values) that result from plugging in all the possible input values (x-values) defined by the domain.
For our function \( f(x) = 3.4 \ln x - 0.25 e^x \), it's essential to analyse the behavior of the terms as \( x \) changes:
  • As \( x \) approaches zero from the positive side, \( \ln x \) decreases sharply to negative infinity while \( e^x \) stays near zero.
  • As \( x \) grows large, the exponential term \( 0.25 e^x \) increases very rapidly and will dominate the logarithmic term \( 3.4 \ln x \).
The function output ranges from negative infinity to positive infinity as the input progresses from just above zero to very large numbers. Therefore, the range of this function is all real numbers \( (-\infty, \infty) \).
Graphing Functions
Graphing functions is an invaluable tool for visualising the behavior of mathematical expressions. For our function, setting up an appropriate window is critical:
  • We suggest using a window with \( x \) ranging from 0.1 to 5, avoiding the undefined region of \( x \leq 0 \).
  • For the \( y \)-axis, starting with a range from -5 to 5 can help capture the essential features of the function. Adjust as needed based on the observed behavior.
With a graphing tool or calculator, plot several points within this window. Carefully examine how the function behaves as \( x \) approaches the window edges. This ensures a complete and accurate depiction of the function's shape and characteristics.
Exponential Functions
Exponential functions are of the form \( e^x \), and they exhibit rapid growth as their input values increase. They have some notable properties:
  • They are always positive, no matter the value of x.
  • They increase dramatically as x becomes positive.
  • When x is negative, they shrink towards zero.
In our function \( f(x) = 3.4 \ln x - 0.25 e^x \), the exponential part \( 0.25 e^x \) greatly influences the result as \( x \) increases. While it's small for smaller values of \( x \), it eventually overshadows the logarithmic term and drives the function to negative infinity.
Natural Logarithm
The natural logarithm, denoted by \( \ln x \), is the inverse of the exponential function. It's defined only for positive values of \( x \) and has these characteristics:
  • It grows very slowly compared to exponential functions.
  • As \( x \) approaches zero from the right, \( \ln x \) drops to negative infinity.
  • As \( x \) increases, \( \ln x \) continues to grow, but at a decreasing rate.
In our function \( f(x) = 3.4 \ln x - 0.25 e^x \), the logarithmic term \( 3.4 \ln x \) significantly impacts the function for small values of \( x \). However, this impact diminishes as \( x \) increases since the exponential term eventually dominates.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The Richter scale, developed in \(1935,\) has been used for years to measure earthquake magnitude. The Richter magnitude \(m\) of an earthquake is given by $$ m=\log \frac{A}{A_{0}} $$ where \(A\) is the maximum amplitude of the earthquake and \(A_{0}\) is a constant. What is the magnitude on the Richter scale of an earthquake with an amplitude that is a million times \(A_{0} ?\)

Home Construction. The chemical urea formaldehyde was used in some insulation in houses built during the mid to late 1960 s. Unknown at the time was the fact that urea formaldehyde emitted toxic fumes as it decayed. The half-life of urea formaldehyde is 1 year. a) What is its decay rate? b) How long will it take \(95 \%\) of the urea formaldehyde present to decay?

As of May 2016 , the highest price paid for a painting was 300 million dollars, paid in 2015 for Willem de Kooning's "Interchange." The same painting was purchased for 20.6 million dollars in \(1989 .\) Data: wsj. com, \(2 / 25 / 16\) a) Find the exponential growth rate \(k,\) and determine the exponential growth function that can be used to estimate the painting's value \(V(t),\) in millions of dollars, \(t\) years after \(1989 .\) b) Estimate the value of the painting in 2025 c) What is the doubling time for the value of the painting? d) How many years after 1989 will it take for the value of the painting to reach 1 billion dollars?

Solve $$ \log _{3} x=-2 $$

Find each of the following, given that $$ f(x)=\frac{1}{x+2} \quad \text { and } \quad g(x)=5 x-8 $$ $$ g g(x) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.