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Chapter 2: Problem 66

Determine whether each function is even, odd, or neither. $$h(x)=2 x^{2}+x^{4}$$

Short Answer

Expert verified
The function \(h(x) = 2x^2 + x^4\) is an even function.

Step by step solution

01

Definition of an Even Function

A function \(f(x)\) is called even if for every \(x\) in the function's domain, \(f(x) = f(-x)\). In other words, replacing every instance of \(x\) with \(-x\) in the function and simplifying should yield the original function.
02

Check if \(h(x)\) is Even

Substitute \(-x\) for \(x\) in \(h(x) = 2x^{2} + x^{4}\) to get \(h(-x) = 2(-x)^{2} + (-x)^{4}\). Simplifying, you get \(h(-x) = 2x^2 + x^4\), which is the equivalent to the original function \(h(x)\). Therefore, \(h(x)\) is an even function.
03

Definition of an Odd Function

A function \(f(x)\) is called odd if for every \(x\) in the function's domain, \(f(x) = -f(-x)\). In other words, replacing every instance of \(x\) with \(-x\) in the function and simplifying should yield the negative of the original function.
04

Check if \(h(x)\) is Odd

For completeness, check if \(h(x)\) may also be odd. Substitute \(-x\) for \(x\) in \(h(x) = 2x^{2} + x^{4}\) to get \(h(-x) = -(2x^2 + x^4)\). But this is not equivalent to the original function \(h(x)\), so \(h(x)\) is not an odd function.

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Most popular questions from this chapter

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. On the other hand, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{3}-2 $$

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 one advantage of using \(f(x)\) rather than \(y\) in a function's equation. What is a piecewise function?

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