Which of the following real-valued functions | vedclass

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Question

(A) function $f(x)$ is even if $f(-x) = f(x)$.$(a)$ $f(x) = x^{3} \sin x$. Then $f(-x) = (-x)^{3} \sin(-x) = (-x^{3})(-\sin x) = x^{3} \sin x = f(x)$.

Vedclass · Accepted Answer

$f(x) = x^{3} \sin x$

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(A) function $f(x)$ is even if $f(-x) = f(x)$.$(a)$ $f(x) = x^{3} \sin x$. Then $f(-x) = (-x)^{3} \sin(-x) = (-x^{3})(-\sin x) = x^{3} \sin x = f(x)$. Thus, $f(x)$ is even.$(b)$ $f(x) = x^{3} \cos x$. Then $f(-x) = (-x)^{3} \cos(-x) = -x^{3} \cos x = -f(x)$. Thus, $f(x)$ is odd.$(c)$ $f(x) = e^{x} x^{3} \sin x$. Then $f(-x) = e^{-x} (-x)^{3} \sin(-x) = e^{-x} x^{3} \sin x 
eq f(x)$. Thus, $f(x)$ is neither even nor odd.$(d)$ $f(x) = x - [x]$. Then $f(-x) = -x - [-x]$. For non-integer $x$, $[-x] = -[x] - 1$, so $f(-x) = -x - (-[x] - 1) = 1 - (x - [x]) = 1 - f(x) 
eq f(x)$. Thus, $f(x)$ is not even.Therefore, option $(a)$ is the correct answer.

Which of the following real-valued functions is an even function?

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