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Chapter 1: Problem 32

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$y^{5}=x^{4}+2$$

Short Answer

Expert verified
The graph of the equation \(y^{5} = x^{4} + 2\) is symmetric with respect to the y-axis only.

Step by step solution

01

Title: Test for symmetry with respect to the y-axis

Replace \(x\) by \(-x\) in the equation to get \((-x)^{4} + 2\). This simplifies to \(x^{4} + 2\), which is the original equation. Hence, \(y^{5} = x^{4} + 2\) is symmetric with respect to the y-axis.
02

Title: Test for symmetry with respect to the x-axis

Replace \(y\) by \(-y\) in the equation to get \((-y)^{5} = x^{4} + 2\). This simplifies to \(-y^{5} = x^{4} + 2\), which is not the original equation. Hence, \(y^{5} = x^{4} + 2\) is not symmetric with respect to the x-axis.
03

Title: Test for symmetry with respect to the origin

Replace \(x\) by \(-x\) and \(y\) by \(-y\) in the equation to get \((-y)^{5} = (-x)^{4} + 2\). This simplifies to \(-y^{5} = x^{4} + 2\), which is not the original equation. Hence, \(y^{5} = x^{4} + 2\) is not symmetric with respect to the origin.

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