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Chapter 4: Problem 4

Find all real numbers (if any) that are fixed points for the given functions. $$F(x)=(7-2 x) / 8$$

Short Answer

Expert verified
The fixed point is \( x = \frac{7}{10} \).

Step by step solution

01

Understand Fixed Points

A fixed point of a function is a value of \( x \) such that \( F(x) = x \). In other words, when substituted into the function, the input equals the output.
02

Set the Equation

To find the fixed point, set the function equal to \( x \): \[ x = \frac{7 - 2x}{8} \]
03

Eliminate the Denominator

Multiply every term by 8 to eliminate the fraction: \[ 8x = 7 - 2x \]
04

Solve for x

First, add \( 2x \) to both sides of the equation: \[ 8x + 2x = 7 \]This simplifies to: \[ 10x = 7 \]Now, divide both sides by 10: \[ x = \frac{7}{10} \]
05

Verify the Solution

Substitute \( x = \frac{7}{10} \) back into the function to verify it is a fixed point: \[ F\left(\frac{7}{10}\right) = \frac{7 - 2\left(\frac{7}{10}\right)}{8} \]Simplify the expression: \[ F\left(\frac{7}{10}\right) = \frac{7 - \frac{14}{10}}{8} \]\[ F\left(\frac{7}{10}\right) = \frac{7 - 1.4}{8} = \frac{5.6}{8} = \frac{7}{10} \]Since \( F\left(\frac{7}{10}\right) = \frac{7}{10} \), the solution \( x = \frac{7}{10} \) is verified as a fixed point.

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Key Concepts

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

Real Numbers
Real numbers are the backbone of most mathematical concepts and are essential when solving equations and functions. They include all the numbers on the number line, which means they comprise:
  • Whole numbers like 0, 1, 2, 3...
  • Integers, which include negative numbers like -1, -2...
  • Fractions and decimals, such as 1/2 or 0.5
  • Irrational numbers like \( \sqrt{2} \) or \( \pi \)

They are characterized by their ability to be represented in decimal form, whether terminating or non-terminating. In the context of functions, finding a fixed point among real numbers implies identifying a specific number that satisfies a particular condition of the function.
Function Equality
Function equality involves understanding when two expressions are equivalent. In the context of finding a fixed point, you set the function equal to the input value:
  • When \( F(x) = x \), it signals that at this particular point, the output of the function matches the input value, characterizing it as a fixed point.

This concept simplifies the search for fixed points by transforming the problem into solving an equation, where the function itself acts as one side of the equation, and the original input is on the other.
Solving Equations
Solving equations is a critical mathematical skill used to find unknown values that satisfy the given mathematical relationships. To find fixed points in a function, you'll need to:
  • Set the equation: We start by equating the function to the input, \( x = \frac{7 - 2x}{8} \).
  • Simplify the equation: Remove any fractions or complex terms, often by multiplying through by the denominator to simplify.
  • Isolate the variable: Rearrange the equation by performing operations such as addition, subtraction, multiplication, or division.

In this case, these steps lead to the solution \( x = \frac{7}{10} \), revealing that this is the fixed point of the given function.
Mathematical Verification
After deriving a solution from solving an equation, it's essential to verify it. Verification ensures that the solution truly satisfies the original problem.
  • Substitute the solution back into the original function to see if both sides are indeed equal.
  • In this exercise, inserting \( x = \frac{7}{10} \) into the function \( F(x) \) results in \[ F\left(\frac{7}{10}\right) = \frac{7 - 2 \left(\frac{7}{10}\right)}{8} = \frac{7}{10} \].
  • Since the function equals the original input, it confirms that \( x = \frac{7}{10} \) is indeed a valid fixed point.

Verification is crucial as it acts like a double-check, preventing errors and confirming the correctness of your results.

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Most popular questions from this chapter

Find quadratic functions satisfying the given conditions. The axis of symmetry is the line \(x=1 .\) The \(y\) -intercept is 1\. There is only one \(x\) -intercept.

(a) Is this a quadratic function? Use a graphing utility to draw the graph. (b) How many turning points are there within the given interval? (c) On the given interval, does the function have a maximum value? A minimum value? $$D(x)=\sqrt{x^{2}-x+1}, \quad x \geq 0 \text { (from Example } 3)$$

(a) An open-top box is to be constructed from a 6 -by- 8 -in. rectangular sheet of tin by cutting out equal squares at each corner and then folding up the resulting flaps. Let \(x\) denote the length of the side of each cutout square. Show that the volume \(V(x)\) is $$V(x)=x(6-2 x)(8-2 x)$$ (b) What is the domain of the volume function in part (a)? [The answer is not \((-\infty, \infty) .]\) (c) Use a graphing utility to graph the volume function, taking into account your answer in part (b). (d) By zooming in on the turning point, estimate to the nearest one-hundredth the maximum volume.

(a) Use a graphing utility to draw a graph of each function. (b) For each \(x\) -intercept, zoom in until you can estimate it accurately to the nearest one-tenth. (c) Use algebra to determine each \(x\) -intercept. If an intercept involves a radical, give that answer as well as a calculator approximation rounded to three decimal places. Check to see that your results are consistent with the graphical estimates obtained in part (b). $$W(u)=2 u^{4}-17 u^{2}+35$$

(a) Use a graphing utility to draw a graph of each function. (b) For each \(x\) -intercept, zoom in until you can estimate it accurately to the nearest one-tenth. (c) Use algebra to determine each \(x\) -intercept. If an intercept involves a radical, give that answer as well as a calculator approximation rounded to three decimal places. Check to see that your results are consistent with the graphical estimates obtained in part (b). $$N(t)=t^{7}+8 t^{4}+16 t$$
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