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Chapter 0: Problem 35

Testing for Symmetry In Exercises \(27-38,\) test for symmetry with respect to each axis and to the origin. $$ y=\frac{x}{x^{2}+1} $$

Short Answer

Expert verified
The function \(y=\frac{x}{x^{2}+1}\) is symmetric with respect to the origin and not symmetric with respect to both the x-axis and y-axis.

Step by step solution

01

Testing Symmetry with respect to Y-axis

The symmetry of a function with respect to the y-axis can be tested by replacing \(x\) with \(-x\) in the function. So, replace \(x\) with \(-x\), so we have, \[y=\frac{-x}{(-x)^{2}+1} = \frac{-x}{x^{2}+1}\] As we can see, this is not the same as the original function, so the function is not symmetric with respect to the y-axis.
02

Testing Symmetry with respect to X-axis

The symmetry of a function with respect to the x-axis can be tested by replacing \(y\) with \(-y\) in the function. This replacement gives us: \[-y=\frac{x}{x^{2}+1}\] Multiplying both sides by -1, yields \[y=-\frac{x}{x^{2}+1}\] which is not the same as the original function, hence the function is not symmetric with respect to the x-axis.
03

Testing Symmetry with respect to Origin

The symmetry of a function with respect to the origin can be tested by replacing \(x\) with \(-x\) and \(y\) with \(-y\). After performing these replacements in the original function we get: \[-y=\frac{-x}{(-x)^{2}+1}= \frac{-x}{x^{2}+1}\] Multiplying both sides by -1 gives: \[y=\frac{x}{x^{2}+1}\] This is the same as the original function, so the function is symmetric with respect to the origin.

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Key Concepts

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

y-axis symmetry
To determine if a function has y-axis symmetry, we examine if it mirrors itself across the y-axis. This essentially involves replacing every instance of the variable \(x\) with \(-x\). If the resulting equation is equivalent to the original function, then y-axis symmetry is confirmed.
In the provided exercise, we consider the function: \[y = \frac{x}{x^2+1}\]. After replacing \(x\) with \(-x\), we have: \[y = \frac{-x}{(-x)^2+1} = \frac{-x}{x^2+1}\].
Comparing this to the original function, it becomes clear that the expressions are not identical. Thus, the function does not exhibit y-axis symmetry.
**Key Points**:
  • Y-axis symmetry implies a reflection over the y-axis.
  • If \(f(x) = f(-x)\), the function is symmetric about the y-axis.
  • In our case, since \(\frac{x}{x^2+1} eq \frac{-x}{x^2+1}\), there's no symmetry around the y-axis.
x-axis symmetry
X-axis symmetry in a function means it reflects along the x-axis. To check for this, change \(y\) to \(-y\) in the equation. If the resulting expression remains consistent with the original equation, the function is symmetric about the x-axis.
Applying this to the function given: Replace \(y\) with \(-y\), leading to: \[-y = \frac{x}{x^2+1}\]. By multiplying both sides by -1, we solve for \(y\) and get: \[y = -\frac{x}{x^2+1}\].
Clearly, this is not the same as the initial expression \(y = \frac{x}{x^2+1}\). Therefore, the function does not have symmetry with respect to the x-axis.
**Important Details**:
  • X-axis symmetry involves a reflection across the x-axis.
  • If modifying \(y\) results in the same function, x-axis symmetry is present.
  • Since \(-\frac{x}{x^2+1}\) does not equal \(\frac{x}{x^2+1}\), there is no symmetry.
origin symmetry
Origin symmetry in a function indicates that rotating the graph by 180 degrees around the origin leads to the same graph. To check for origin symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\). If the transformed equation matches the original form, such symmetry exists.
For our function, initiate this verification by substituting both \(x\) and \(y\): \[-y = \frac{-x}{(-x)^2+1} = \frac{-x}{x^2+1}\]. Solving for \(y\) by multiplying each side by -1 results in: \[y = \frac{x}{x^2+1}\].
Seeing that this matches exactly with the original equation indicates that the function is indeed symmetric about the origin.
**Key Aspects**:
  • Origin symmetry denotes rotational symmetry by 180 degrees around the origin.
  • If \(-f(x) = f(-x)\), the function exhibits origin symmetry.
  • In our scenario, since \(\frac{x}{x^2+1} = \frac{x}{x^2+1}\), origin symmetry is confirmed.

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