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Chapter 9: Problem 2

Create a table of values for each function for values near its non-permissible value. Explain how your table shows whether a point of discontinuity or an asymptote occurs in each case. a) \(y=\frac{x^{2}-3 x}{x}\) b) \(y=\frac{x^{2}-3 x-10}{x-2}\) c) \(y=\frac{3 x^{2}+4 x-4}{x+4}\) d) \(y=\frac{5 x^{2}+4 x-1}{5 x-1}\)

Short Answer

Expert verified
a) Discontinuity at x = 0. b) Asymptote at x = 2.

Step by step solution

01

Define the Non-Permissible Values

Find the values of x where each function is not defined. These are the values where the denominator is zero.
02

Simplify Each Function

Simplify the functions to see if any common factors in the numerator and denominator can be canceled out. This helps in identifying if the point is a discontinuity or an asymptote.
03

Create a Table of Values Near the Non-Permissible Value

Generate values close to the non-permissible value to see the behavior of the function near these values.
04

Identify Discontinuity or Asymptote

Analyze the table of values and simplified function to determine if the non-permissible value causes a hole (discontinuity) or a vertical asymptote.
05

Example Solution: Part a)

For part (a): - Function: \( y = \frac{x^{2} - 3x}{x} \) - Simplify: \( y = x - 3 \) for all \( x eq 0 \)- Non-Permissible Value (NPV): \( x = 0 \)- Table of Values:\[\begin{array}{ccc} x & y = \frac{x^{2} - 3x}{x} & y = x-3 \-0.1 & -3.1 & -3.1 \-0.01 & -3.01 & -3.01 \0.01 & -2.99 & -2.99 \0.1 & -2.9 & -2.9 \end{array}\]- Conclusion: The function simplifies to \(y = x - 3\) with a hole (discontinuity) at \(x = 0\).
06

Example Solution: Part b)

For part (b): - Function: \( y = \frac{x^{2} - 3x - 10}{x - 2} \) - Factor Numerator: \( x^{2} - 3x - 10 = (x - 5)(x + 2) \) - Simplify: \( y = \frac{(x - 5)(x + 2)}{x - 2} \) - NPV: \( x = 2 \)- Table of Values:\[\begin{array}{ccc} x & y = \frac{x^{2} - 3x - 10}{x-2} & Simplified \1.9 & -5.0 & -2.7 \1.99 & -5.0 & -2.98 \2.01 & -5.0 & -3.02 \2.1 & -5.0 & -3.25 \end{array}\]- Conclusion: There is a vertical asymptote at \(x = 2\).

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Key Concepts

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

Non-Permissible Values
Non-permissible values are values of the variable that make the denominator of a function zero, rendering the function undefined at those points.
To find non-permissible values, set the denominator equal to zero and solve for the variable. For example, in the function \( \frac{3x}{x-2} \), the non-permissible value is \( x = 2 \) because the denominator becomes zero when \( x = 2 \).
Non-permissible values help determine if a function is discontinuous or has a vertical asymptote.
Simplification of Functions
Simplifying functions involves canceling common factors in the numerator and denominator.
For example, consider the function \( \frac{x^2 - 3x}{x} \). The numerator can be factored to \( x(x - 3) \), leading to simplification: \( \frac{x(x-3)}{x} = x - 3 \) for \( x eq 0 \).
This simplification reveals the function鈥檚 behavior and helps identify discontinuities or asymptotes.
If canceling a factor removes a non-permissible value, the function has a discontinuity (hole) there.
Tables of Values
Creating tables of values around non-permissible points helps visualize how a function behaves near these values.
Steps to create a table of values:
  • Identify the non-permissible value.
  • Choose values close to it, both smaller and larger.
  • Calculate the function鈥檚 value at these points.
For example, if \( y = \frac{x^2 - 3x}{x} \) around \( x=0 \), choose values like -0.1, -0.01, 0.01, and 0.1 and calculate \( y \).
This helps uncover trends and determine if there is a discontinuity or asymptote.
Discontinuity
Discontinuity occurs when a function is not defined at a certain point or cannot be made continuous by redefining it.
For instance, \( y = \frac{x^2 - 3x}{x} = x - 3 \) has a discontinuity at \( x = 0 \) because \( y \) is not defined there.
The graph will have a 'hole' at the point of discontinuity.
To confirm discontinuity, check if the function simplifies to a form where the non-permissible value is removable.
Vertical Asymptote
A vertical asymptote occurs when a function approaches infinity as the variable approaches the non-permissible value.
For example, \( y = \frac{x^2 - 3x - 10}{x - 2} \) can be factored and simplified to \( y = \frac{(x-5)(x+2)}{x-2} \), resulting in an asymptote at \( x = 2 \).
As \( x \) approaches 2 from either side, \( y \) will increase or decrease without bound.
Asymptotes indicate that the function's value grows extremely large or small near the non-permissible value.

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Most popular questions from this chapter

a) Determine the roots of the rational equation \(-\frac{2}{x}+x+1=0\) algebraically. b) Graph the rational function \(y=-\frac{2}{x}+x+1\) and determine the \(x\) -intercepts. c) Explain the connection between the roots of the equation and the \(x\) -intercepts of the graph of the function.

Consider the functions \(f(x)=\frac{x+a}{x+b}\) \(g(x)=\frac{x+a}{(x+b)(x+c)},\) and \(h(x)=\frac{(x+a)(x+c)}{(x+b)(x+c)},\) where \(a, b,\) and \(c\) are different real numbers. a) Which pair of functions do you think will have graphs that appear to be most similar to each other? Explain your choice. b) What common characteristics will all three graphs have? Give reasons for your answer.

A convex lens focuses light rays from an object to create an image, as shown in the diagram. The image distance, \(I\), is related to the object distance, \(b,\) by the function \(I=\frac{f b}{b-f},\) where the focal length, \(f,\) is a constant for the particular lens used based on its specific curvature. When the object is placed closer to the lens than the focal length of the lens, an image is perceived to be behind the lens and is called a virtual image. a) Graph \(I\) as a function of \(b\) for a lens with a focal length of \(4 \mathrm{cm} .\) b) How does the location of the image change as the values of \(b\) change? c) What type of behaviour does the graph exhibit for its non-permissible value? How is this connected to the situation?

A truck leaves Regina and drives eastbound. Due to road construction, the truck takes \(2 \mathrm{h}\) to travel the first \(80 \mathrm{km} .\) Once it leaves the construction zone, the truck travels at \(100 \mathrm{km} / \mathrm{h}\) for the rest of the trip. a) Let \(v\) represent the average speed, in kilometres per hour, over the entire trip and \(t\) represent the time, in hours, since leaving the construction zone. Write an equation for \(v\) as a function of \(t\) b) Graph the function for an appropriate domain. c) What are the equations of the asymptotes in this situation? Do they have meaning in this situation? Explain. d) How long will the truck have to drive before its average speed is \(80 \mathrm{km} / \mathrm{h} ?\) e) Suppose your job is to develop GPS technology. How could you use these types of calculations to help travellers save fuel?

Determine the solution to the equation \(\frac{2 x+1}{x-1}=\frac{2}{x+2}-\frac{3}{2}\) using two different methods.
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