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1 6. R (x) = 1- (-2 — х)?

1 6. R (x) = 1- (-2 — х)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Graph the following rational functions using transformations, state the domain, and all asymptotes.

6. \( R(x) = 1 - \frac{1}{(-2-x)^2} \) This function defines \( R(x) \) in terms of \( x \). It consists of a constant value of 1 subtracted by the reciprocal of the square of the expression \((-2-x)\). ### Explanation: - **Numerator:** The numerator of the fraction is 1. - **Denominator:** The denominator is the square of \((-2-x)\). - **Function Behavior:** Since the denominator involves a square, the function's behavior is influenced by whether the expression \((-2-x)\) approaches zero, which would make the function undefined at that value of \( x \). This equation can be analyzed for various values of \( x \) to study its behavior, such as limits, asymptotes, and points of discontinuity.
Transcribed Image Text:6. \( R(x) = 1 - \frac{1}{(-2-x)^2} \) This function defines \( R(x) \) in terms of \( x \). It consists of a constant value of 1 subtracted by the reciprocal of the square of the expression \((-2-x)\). ### Explanation: - **Numerator:** The numerator of the fraction is 1. - **Denominator:** The denominator is the square of \((-2-x)\). - **Function Behavior:** Since the denominator involves a square, the function's behavior is influenced by whether the expression \((-2-x)\) approaches zero, which would make the function undefined at that value of \( x \). This equation can be analyzed for various values of \( x \) to study its behavior, such as limits, asymptotes, and points of discontinuity.
Expert Solution
Step 1

firstly we will draw graph of 1x2

Advanced Math homework question answer, step 1, image 1

Then we know the graph of 1(-2-x)2=1(x+2)2will shift the graph of 1x2 to 2 units left, as shown 

Advanced Math homework question answer, step 1, image 2

Now we will draw the graph of -1(x+2)2 and it will downwards the graph of 1(x+2)2

Advanced Math homework question answer, step 1, image 3

Now we will add 1 to the given graph, we know it will shift our graph to 1 units upwards.

Advanced Math homework question answer, step 1, image 4

which is the required graph.

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