/*! 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 110 In general, it is not possible t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 110

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$3^{-x}=\sqrt{x+5}$$

Short Answer

Expert verified
The solution is approximately \(x = -0.445\).

Step by step solution

01

Understanding the Equation

The equation given is \(3^{-x} = \sqrt{x+5}\). This equation involves an exponential function on the left side and a radical (square root) function on the right side.
02

Setting Up Functions for Graphing

Define two functions from the equation: \( y_1 = 3^{-x} \) and \( y_2 = \sqrt{x+5} \). Our goal is to find the values of \(x\) where the graphs of these two functions intersect.
03

Graphing the Functions

Use graphing software or calculators to plot \( y_1 = 3^{-x} \) and \( y_2 = \sqrt{x+5} \). Note that the domain for \( y_1 \) is all real numbers, but for \( y_2 \), \(x\) must be \( \geq -5 \) because of the square root of \(x+5\).
04

Finding Intersection Points

Identify the points where the two graphs intersect. These intersection points correspond to the solutions of the equation \(3^{-x} = \sqrt{x+5}\).
05

Approximating Solutions

Once you have identified the intersection points, read the \(x\)-coordinates from the graph. Approximating these values to the nearest thousandth gives a solution of \(x \approx -0.445\).

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.

Exponential Functions
Exponential functions are an important concept in mathematics that describe situations of rapid growth or decay. In the given equation, we have the function \( y_1 = 3^{-x} \), showcasing an exponential decay.
  • **General Form**: Exponential functions can be generally expressed as \( a^x \), where \( a \) is a constant.
  • **Behavior**: As \( x \) increases, \( 3^{-x} \) decreases because the negative exponent inverts the power.
  • **Domains and Ranges**: The domain is all real numbers, and the range is positive real numbers but bounded between 0 and 1 for \( 3^{-x} \).
Understanding exponential functions helps in analyzing phenomena such as population growth, radioactive decay, and, as in our problem, intersections with other functions.
Radical Functions
Radical functions involve roots, and in our equation, the function \( y_2 = \sqrt{x+5} \) includes a square root.
  • **Significance**: Radical functions help in expressing quantities that vary slowly.
  • **Domains**: As seen from \( \sqrt{x+5} \), the domain must satisfy \( x+5 \geq 0 \), hence \( x \geq -5 \).
  • **Range**: For our function, the range is all non-negative real numbers, since square roots produce non-negative results.
Plotting this function alongside exponential ones can help find their intersection points visually, as both functions behave differently across their domains.
Intersection Points
Intersection points are where two graphs meet, and they hold the key to solving the given equation graphically.
  • **Determination**: By graphing both \( y_1 \) and \( y_2 \), the intersection points can be visually identified on the graph.
  • **Significance**: The \( x \)-coordinates of these points are solutions to the equation \( 3^{-x} = \sqrt{x+5} \).
  • **Visual Aid**: These points provide a geometric perspective, helping us see solutions that might be complex to solve analytically.
Finding these points can often be achieved using graphing software, calculators, or manually plotting points.
Numerical Approximation
Graphical methods, while effective, often result in approximations rather than precise solutions. Numerical approximation helps refine these solutions.
  • **Approach**: Once the intersection is identified, numerical methods estimate \( x \)-values to required precision.
  • **Precision**: In our problem, this means finding \( x \) to the nearest thousandth, necessary for practical applications.
  • **Tools**: Calculators or software can assist in finely tuning the approximation from the graph.
Numerical approximation is a valuable skill, particularly in real-world problems where an exact solution is not feasible or necessary.

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

Salinity The salinity of the oceans changes with latitude and depth. In the tropics, the salinity increases on the surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$f(x)=31.5+1.1 \log (x+1)$$ models salinity to depths of 1000 meters at a latitude of 57.5". The variable x is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D .\) Global Physical Climatology, Academic Press.) Approximate analytically, to the nearest meter, the depth where the salinity equals 33.

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=3 x-7, \quad g(x)=\frac{x+7}{3}$$

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=\frac{-x}{x-4} ;[-2.6,10.6] \text { by }[-4.1,4.1]$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\log _{b}(2 y+5)-\frac{1}{2} \log _{b}(y+3)$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\frac{1}{2} \log x-\frac{1}{3} \log y-2 \log z$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.