Is the following function even, odd, or neither? $\newline$ $f(x)=\frac{x^{3}}{x+1}$ $\newline$ Choose $1$ answer: $\newline$ (A) Even $\newline$ (B) Odd $\newline$ (C) Neither

Full solution

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

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f(x)=\frac{x^{3}}{x+1}

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Choose

1

answer:

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(A) Even

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(B) Odd

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Check Symmetry Properties: 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, and an odd function satisfies $f(-x) = -f(x)$ for all $x$ in its domain.
Check if $f(x)$ is Even: First, we will check if $f(x)$ is even. We calculate $f(-x)$ and see if it is equal to $f(x)$ . $\newline$ $f(-x) = ((-x)^3)/((-x)+1) = (-x^3)/(-x+1) = -x^3/(1-x)$
Compare $f(-x)$ with $f(x)$ : Now we compare $f(-x)$ with $f(x)$ . $f(x) = \frac{x^3}{x+1}$ $f(-x) = \frac{-x^3}{1-x}$ We can see that $f(-x)$ is not equal to $f(x)$ , because $f(x)$ has $x+1$ in the denominator, while $f(-x)$ has $f(x)$ $1$ (which is $f(x)$ $2$ (x+ $1$ )). Therefore, $f(x)$ is not even.
Check if $f(x)$ is Odd: Next, we will check if $f(x)$ is odd. We calculate $f(-x)$ and see if it is equal to $-f(x)$ . We already have $f(-x) = -\frac{x^3}{1-x}$ Now we calculate $-f(x) = -\left(\frac{x^3}{x+1}\right) = -\frac{x^3}{x+1}$
Compare $f(-x)$ with $-f(x)$ : We compare $f(-x)$ with $-f(x)$ .
$f(-x) = \frac{-x^3}{1-x}$
$-f(x) = \frac{-x^3}{x+1}$
We can see that $f(-x)$ is not equal to $-f(x)$ , because the denominators are different: $(1-x)$ is not the same as $(x+1)$ . Therefore, $-f(x)$ $0$ is not odd.
Conclusion: Since $f(x)$ is neither even nor odd, the correct answer is $(C)$ Neither.