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Chapter 0: Problem 70

Use the algebraic tests to check for symmetry with respect to both axes and the origin. \(y=\frac{1}{x^2+1}\)

Short Answer

Expert verified
The function \(y=\frac{1}{x^2+1}\) is symmetric with respect to the y-axis, but it is neither symmetric with respect to the x-axis nor the origin.

Step by step solution

01

Check for symmetry with respect to the y-axis

Replace \(x\) by \(-x\) in the equation and simplify. If the equation remains the same, the function is symmetric with respect to the y-axis. Thus check if \(y=\frac{1}{(-x)^2+1}\) simplifies to \(y=\frac{1}{x^2+1}\). In this case, it does, indicating y-axis symmetry.
02

Check for symmetry with respect to the x-axis

Replace \(y\) with \(-y\) and solve the equation. If you wind up with the original equation, the function is symmetric with respect to the x-axis. In this case \(-y=\frac{1}{x^2+1}\) and solving for \(y\) leads to \(y=-\frac{1}{x^2+1}\). This does not resemble the original equation, thus the function has no x-axis symmetry.
03

Check for symmetry with respect to the origin

To check for origin symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation and simplify to see if the original equation results. For this function \(-y=\frac{1}{(-x)^2+1}\) which simplifies to \(y=-\frac{1}{x^2+1}\). This does not resemble the original equation thus, no symmetry about 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
A function is said to have y-axis symmetry if, when you replace every instance of \(x\) with \(-x\), the equation remains unchanged. This means that the right side of the graph mirrors exactly the left side over the y-axis. For our function, \(y = \frac{1}{x^2 + 1}\), replacing \(x\) with \(-x\) gives us \(y = \frac{1}{(-x)^2 + 1} = \frac{1}{x^2 + 1}\). Since both versions of the function are identical, the original function is symmetric with respect to the y-axis.
  • This type of symmetry is like looking at yourself in a mirror lying flat, the image is exactly identical.
  • A practical example would be the function \(y = x^2\), which has y-axis symmetry.
x-axis symmetry
For a function to have x-axis symmetry, replacing \(y\) by \(-y\) in the equation should still yield the original function. If this is the case, it means that the graph below the x-axis mirrors the part above the x-axis. However, for \(y = \frac{1}{x^2 + 1}\), substituting \(-y\) for \(y\), you get \(-y = \frac{1}{x^2 + 1}\). Solving this does not return to the original equation; instead, it results in \(y = -\frac{1}{x^2 + 1}\), showing that the function does not have x-axis symmetry.
  • Imagine folding a piece of paper along the x-axis; x-axis symmetry would mean both halves align perfectly.
  • Not all functions exhibit this type of symmetry, examples include circles centered on the origin like \(x^2 + y^2 = 1\).
origin symmetry
A function may have origin symmetry if replacing both \(x\) with \(-x\) and \(y\) with \(-y\) returns the original equation. This type of symmetry means the graph is identical when rotated 180 degrees around the origin. Let's check this for \(y = \frac{1}{x^2 + 1}\). Substituting both \(x\) with \(-x\) and \(y\) with \(-y\), we get \(-y = \frac{1}{(-x)^2 + 1}\), which simplifies to \(y = -\frac{1}{x^2 + 1}\). This does not match the original equation, indicating the function is not symmetric about the origin.
  • Origin symmetry is like spinning an old winch, where everything looks the same after a complete rotation.
  • Every odd function exhibits this type of symmetry, an example is \(y = x^3\).

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ q(x)=(x-5)^{2} $$

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ f^{-1}\right)(-3) $$

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(x)=x^{6}-2 x^{2}+3 $$

True or False? Determine whether the statement is true or false. Justify your answer. If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)
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