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Chapter 2: Problem 64

Determine whether each function is even, odd, or neither. $$g(x)=x^{2}-x$$

Short Answer

Expert verified
The function \(g(x)=x^{2}-x\) is neither even nor odd.

Step by step solution

01

Check for 'Even' Property

Firstly, we substitute \(x\) with \(-x\) in the function which yields \(g(-x) = (-x)^{2} - (-x) = x^{2}+x\). This is not equal to \(g(x)\), so the function is not even.
02

Check for 'Odd' Property

Now, consider if our function is an odd function. We require \(g(-x) = -g(x)\). From 'Step 1', we know that \(g(-x) = x^{2}+x\), and we calculate \(-g(x) = -(x^{2} - x) = -x^{2} + x\). Clearly, \(g(-x) \neq -g(x)\), showing that the function is not odd.
03

Conclusion

As the function fulfills neither the even nor the odd property, it can be concluded that the function \(g(x)=x^{2}-x\) is neither even nor odd.

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Most popular questions from this chapter

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{3}-3 $$

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{2}-2 $$

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)=|x+3| $$

If \(f(x)=a x^{2}+b x+c\) and \(r_{1}=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\) find \(f\left(r_{1}\right)\) without doing any algebra and explain how you arrived at your result.

Describe how to write the equation of a line if two points along the line are known.
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