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Use the following information for next two questions: A function $f(x)$ is said to be even if $f(-x) = f(x)$, and odd if $f(-x) = -f(x)$. Thus, for example, the function given by $f(x)=x^{2}$ is even, while the function given by $f(x)=x^{3}$ is odd. Using this definition, answer the following questions.

The function given by $f(x) = |x|^{3}$

  1. even 
  2. odd 
  3. neither 
  4. both
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Given that, $f(-x) = \left\{\begin{matrix} f(x) ;  & even  \\ -f(-x) ; & odd  \end{matrix}\right.$

$f(x) = |x|^{3} \longrightarrow (1)$

We know that $f(x) = |x| = x $

$\qquad f(-x) = |-x| = x $

So, $f(x) = |x| $ is even function.

Now, $f(x) = |x|^{3}$

$\Rightarrow \boxed{f(x) = x^{3} (odd)}$

Correct Answer $: \text{B}$

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