Problem 35 Is there a real number $x$ suc... [FREE SOLUTION]

91影视

Algebra and Trigonometry with Analytic Geometry

Earl Swokowski, Jeffery A. Cole

$Math Studyset 91影视 Explanations$ Math

12 Edition

Chapter 9: Problem 35

Is there a real number $x$ such that $x=2^{-x}$ ? Decide by displaying graphically the system $$ \left\\{\begin{array}{l} y=x \\ y=2^{-x} \end{array}\right. $$

Short Answer

Expert verified

Yes, there is an intersection point where $x=2^{-x}$, approximately $x = 0.5$.

Step by step solution

Understand the Problem

We need to determine if there is a real number $x$ such that $x=2^{-x}$. This involves finding the intersection point(s) between the line $y=x$ and the curve $y=2^{-x}$.

Sketch the Linear Function

First, we plot the line $y=x$. This is a straight line that passes through the origin (0,0) with a slope of 1, which means for every unit increase in $x$, $y$ also increases by one unit.

Sketch the Exponential Function

Now, plot the curve $y=2^{-x}$. This is an exponential decay function. When $x=0$, $y=2^{0}=1$. As $x$ increases, $y$ approaches zero but never actually reaches it. As $x$ becomes large and negative, $y$ increases sharply.

Find the Intersection Points

To find if $x=2^{-x}$ has a real solution, we look for the intersection of the two graphs. By observing the plots of $y=x$ and $y=2^{-x}$, notice that they intersect at a single point.

Verify the Intersection Point

From the graphical representation, the functions intersect somewhere between $x=0$ and $x=1$. To confirm this, pick a value like $x=0.5$. Substitute in $y=2^{-0.5}$, which gives $0.5=2^{-0.5}$. This holds because $2^{-0.5}=\frac{1}{\sqrt{2}}\approx0.707$. Thus, $x=0.5$ is close to where the intersection occurs.

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Key Concepts

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

Exponential Functions

An exponential function is a mathematical function of the form $y = a^x$, where $a$ is a constant and $x$ is the exponent. In this context, the function $y = 2^{-x}$ is called an exponential decay function. When the exponent is negative, it represents a function that decreases rapidly as $x$ increases.

When $x = 0$, we have $2^{-0} = 1$. This gives us the point (0, 1) on the graph, indicating that the function starts at 1 when $x$ is zero.
As $x$ increases, $y$ rapidly approaches zero, but never quite gets there. Thus, the x-axis serves as a horizontal asymptote.
If $x$ is negative, each increase makes $y$ grow exponentially larger.

Understanding these features helps visualize where it might intersect with other functions, such as linear functions.

Linear Functions

A linear function is simply one that creates a straight line when graphed. The most basic form is $y = mx + b$, with $m$ as the slope and $b$ as the y-intercept. In the problem, the linear function given is $y = x$, which has a slope $m = 1$ and a y-intercept $b = 0$.

This function creates a 45-degree line that passes through the origin (0,0).
For every 1-unit increase in $x$, $y$ increases by the same amount, meaning the slope is 1.
This kind of function is fundamental in mathematics because of its simplicity and predictability.

The line $y = x$ serves as a reference point to explore intersections with more complex functions like exponential functions.

Intersection Points

The intersection point between two functions is where they share the same coordinates on a graph. This problem focuses on finding where the linear function $y = x$ intersects the exponential function $y = 2^{-x}$.

Graphically, these appear as the points where the plotted lines cross each other.
To determine the intersection algebraically, set the two equations equal: $x = 2^{-x}$.
Solving this equation involves using both graphing and algebraic techniques to find approximate or exact solutions.

Visualization is crucial here, as plotting the graphs provides a clear insight into where and if these functions intersect.

Real Solutions

A real solution is a value for $x$ that satisfies both equations in the system with real numbers only. When graphing $y = x$ and $y = 2^{-x}$, we look for intersection points that can be confirmed visually or algebraically.

In the graphical approach, we notice if and where the graphs meet. From the given task, the intersection occurs near $x = 0.5$.
This is verified by substituting into the equations to see if the values hold true, for example, $2^{-0.5}\approx 0.707$.
Real solutions imply those solutions have tangible, positive or negative real-number values, unlike imaginary numbers.

Confirming real solutions gives the assurance that the values are practical and applicable to real-world problems.

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91影视

Is there a real number \(x\) such that \(x=2^{-x}\) ? Decide by displaying graphically the system $$ \left\\{\begin{array}{l} y=x \\ y=2^{-x} \end{array}\right. $$

Short Answer

Step by step solution

Understand the Problem

Sketch the Linear Function

Sketch the Exponential Function

Find the Intersection Points

Verify the Intersection Point

Key Concepts

Exponential Functions

Linear Functions

Intersection Points

Real Solutions

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