Is the following function even, odd, or neither? $\newline$ $f(x)=x^{4}+x$ $\newline$ Choose $1$ answer: $\newline$ (A) Even $\newline$ (B) Odd $\newline$ (C) Neither

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Algebra 2

Symmetry and periodicity of trigonometric functions

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Q. Is the following function even, odd, or neither?

\newline

f(x)=x^{4}+x

\newline

Choose

1

answer:

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(A) Even

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(B) Odd

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Symmetry properties of a function: To determine if the function $f(x)$ is even, odd, or neither, we need to check the symmetry properties of the function. An even function satisfies $f(x) = f(-x)$ for all $x$ in its domain, while an odd function satisfies $f(-x) = -f(x)$ for all $x$ in its domain.
Checking if f(x) is even: Let's first check if f(x) is even. We substitute $-x$ for $x$ in the function and see if we get the original function back. $f(-x) = (-x)^{4} + (-x) = x^{4} - x$
Checking if $f(x)$ is odd: We can see that $f(-x) \neq f(x)$ , because $f(x) = x^{4} + x$ and $f(-x) = x^{4} - x$ . Therefore, the function $f(x)$ is not even.
Conclusion: Next, let's check if $f(x)$ is odd. We substitute $-x$ for $x$ in the function and see if we get the negative of the original function. $\newline$ $f(-x) = (-x)^{4} - x = x^{4} - x$
Conclusion: Next, let's check if $f(x)$ is odd. We substitute $-x$ for $x$ in the function and see if we get the negative of the original function. $\newline$ $f(-x) = (-x)^{4} - x = x^{4} - x$ We can see that $f(-x)$ does not equal $-f(x)$ , because $-f(x)$ would be $-x^{4} - x$ , which is not equal to $x^{4} - x$ . Therefore, the function $f(x)$ is not odd.
Conclusion: Next, let's check if $f(x)$ is odd. We substitute $-x$ for $x$ in the function and see if we get the negative of the original function. $f(-x) = (-x)^{4} - x = x^{4} - x$ We can see that $f(-x)$ does not equal $-f(x)$ , because $-f(x)$ would be $-x^{4} - x$ , which is not equal to $x^{4} - x$ . Therefore, the function $f(x)$ is not odd.Since $f(x)$ is neither even nor odd, the correct answer is that the function is neither even nor odd.