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Chapter 4: Problem 23

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{1}{x-1}+1$$

Short Answer

Expert verified
The graph is a hyperbola shifted 1 unit right and up; domain is \( x \neq 1 \) and range is \( y \neq 1 \).

Step by step solution

01

Identify the parent function

The function given is a transformation of the parent function \( y = \frac{1}{x} \). The term \( \frac{1}{x-1} \) indicates this is based on \( \frac{1}{x} \) as it is not squared.
02

Identify the transformations

The function \( f(x) = \frac{1}{x-1} + 1 \) includes two transformations:1. Horizontal shift: The \( x-1 \) inside the fraction indicates a shift to the right by 1 unit.2. Vertical shift: The \( +1 \) outside the fraction indicates a shift upward by 1 unit.
03

Sketch the transformed graph

Start with the graph of \( y = \frac{1}{x} \), which has vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \) respectively. Shift the graph 1 unit to the right and 1 unit up. The new asymptotes are at \( x = 1 \) and \( y = 1 \). Draw the hyperbola approaching these new asymptotes.
04

Create an accurate graph using a calculator

Input the function \( f(x) = \frac{1}{x-1} + 1 \) into a graphing calculator to generate a precise graph. Use the calculator to verify the transformations and observe the location of the asymptotes.
05

Determine the domain and range

The domain of \( f(x) = \frac{1}{x-1} + 1 \) includes all real numbers except \( x = 1 \), where there is a vertical asymptote. Therefore, the domain is \( x eq 1 \). The range is all real numbers except \( y = 1 \), due to the horizontal asymptote. Thus, the range is \( y eq 1 \).

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Key Concepts

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

Parent Function
A parent function is the simplest form of a function type. For rational functions, a classic example is the function \( y = \frac{1}{x} \). This function forms a hyperbola with characteristic curves in different quadrants. Recognizing the parent function is crucial because it helps to understand the foundation before applying transformations. A transformation changes the graph's position or shape without altering the basic properties of the function.
  • For \( y = \frac{1}{x} \), the graph has two branches, one in the first quadrant and another in the third quadrant relative to the origin.
  • Key features of the parent function include asymptotes, which are lines the graph approaches but never touches.
Understanding the parent function allows you to predict the behavior of transformed graphs and is an essential skill in graph analysis.
Asymptotes
Asymptotes are lines that a graph gets closer to but never actually reaches. They are crucial in understanding the behavior and limits of rational functions. There are generally two types of asymptotes: vertical and horizontal.
  • Vertical Asymptotes: These occur at values of \( x \) that make the function undefined. In our example function \( f(x) = \frac{1}{x-1} + 1 \), the vertical asymptote is at \( x = 1 \), because the function cannot take a value where the denominator is zero.
  • Horizontal Asymptotes: These describe the behavior of the function as \( x \) approaches infinity. For our function, the horizontal asymptote occurs at \( y = 1 \), reflecting the constant term added to the parent function.
Identifying these lines is crucial as they define the bounds within which the graph lies and indicate how the graph stretches infinitely along them.
Domain and Range
Domain and range describe the set of possible input and output values for a function, respectively. For the function \( f(x) = \frac{1}{x-1} + 1 \), understanding the domain and range provides insight into the graph's behavior.
  • Domain: This refers to all possible \( x \) values for which the function is defined. Since the function becomes undefined at \( x = 1 \) (due to division by zero), the domain excludes \( x = 1 \). Thus, the domain is all real numbers except \( x = 1 \), or \( x eq 1 \).
  • Range: The range represents all possible output \( y \) values. The horizontal asymptote at \( y = 1 \) implies that the function never actually reaches this value, though it comes very close. Therefore, the range is all real numbers except \( y = 1 \), or \( y eq 1 \).
Knowing the domain and range helps in comprehending the function's graph and its limitations.
Graphing Calculator
A graphing calculator is a powerful tool for visualizing functions and validating graph transformations. When dealing with complex functions, such as \( f(x) = \frac{1}{x-1} + 1 \), the graphing calculator allows you to see the immediate effect of each transformation in action.
  • Inputting the Function: By entering \( f(x) \) into the graphing calculator, you can quickly generate an accurate depiction of the graph. This helps you to verify and understand the transformation of the parent function.
  • Observing Asymptotes: The calculator's output will highlight where the graph approaches its asymptotes. This visualization reinforces your understanding of how the transformations affect the graph.
  • Check Transformations: Observing the graph on a calculator lets you confirm the horizontal and vertical shifts. It provides a clear picture that supports your analytical work done by hand.
Having a graphing calculator at hand provides a visual check against your manual graphing, ensuring your transformations and calculations are accurate.

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Most popular questions from this chapter

Solve each problem. Wing Size Suppose that the surface area \(S\) of a bird's wings, in square feet, can be modeled by $$ S(w)=1.27 w^{2 / 3} $$ where \(w\) is the weight of the bird in pounds. Estimate the surface area of a bird's wings if the bird weighs 4.0 pounds.

Solve each problem. Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per minute, while the average number of vehicles being admitted through the gate per minute is equal to \(x .\) Then the average waiting time in minutes for each vehicle at the gate is given by $$f(x)=\frac{x-5}{x^{2}-10 x}$$ where \(x>10 .\) (Source: Mannering, \(\mathrm{F}\). and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, 2d. ed., John Wiley and Sons.) (a) Estimate the admittance rate \(x\) that results in an average wait of 15 seconds. (b) If one attendant can serve 5 vehicles per minute, how many attendants are needed to keep the average wait to 15 seconds or less?

Solve each rational inequality by hand. Do not use a calculator. $$\frac{x(x-3)}{x+2} \geq 0$$

Solve each problem involving rate of work. Mrs. Schmulen is a high school mathematics teacher. She can grade a set of chapter tests in 5 hours working alone. If her student teacher Elwyn helps her, it will take 3 hours to grade the tests. How long would it take Elwyn to grade the tests if he worked alone?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=2 y^{2}$$
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