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Chapter 1: Problem 35

Use the algebraic tests to check for symmetry with respect to both axes and the origin. $$y=x^{3}$$

Short Answer

Expert verified
The function \(y=x^{3}\) is symmetric with respect to the origin, but not with respect to the y-axis. The test for symmetry with respect to the x-axis is not applicable to this function as functions by definition can't be symmetric about the x-axis.

Step by step solution

01

Symmetry with Respect to the y-axis (Even Function Test)

Replace \(x\) with \(-x\) in the equation. The original equation is \(y=x^{3}\). Let's substitute \(-x\) for \(x\): \(y=(-x)^{3}\) Simplifying gives \(y=-x^3\). This isn't the same as the original equation, so the function is not symmetric about the y-axis.
02

Symmetry with Respect to the Origin (Odd Function Test)

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. The original equation is \(y=x^{3}\). Let's substitute \(-x\) for \(x\) and \(-y\) for \(y\): \(-y=(-x)^{3}\). Simplifying gives \(-y=-x^3\), which is the same as the original equation when we multiply through by -1. So the function is symmetric with respect to the origin.
03

Symmetry with Respect to the x-axis

This test is not applicable to most functions (including this one) since functions by definition only have one output (\(y\) value) for each input (\(x\) value)

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Most popular questions from this chapter

Evaluate the function for the indicated values. \(h(x)=[x+3]\) (a) \(h(-2)\) (b) \(h\left(\frac{1}{2}\right)\) (c) \(h(4.2)\) (d) \(h(-21.6)\)

Find the difference quotient and simplify your Answer: $$f(x)=x^{2 / 3}+1, \quad \frac{f(x)-f(8)}{x-8}, \quad x \neq 8$$

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Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(z\) varies jointly as \(x\) and \(y .(z=64 \text { when } x=4\) and \(y=8 .)\)

Find the difference quotient and simplify your Answer: $$f(x)=x^{2}-x+1, \quad \frac{f(2+h)-f(2)}{h}, \quad h \neq 0$$
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