Question

Given $x$ is real, identify all the even-functions among the following:

• A. $x|x|$
• B. $\dfrac{\cos(x)}{x}$
• C. $\sin(x^2)$
• D. $e^{-|x|}$

Hint:

Even functions remain unchanged when $x$ is replaced by $-x$.

Answer 1

The Correct Option is C, D

Step 1: Definition of even function.
A function $f(x)$ is even if $f(-x) = f(x)$ for all real $x$.
Step 2: Check each option.
(A) $x|x|$ is odd because $(-x)|-x| = -x|x| \neq x|x|$.
(B) $\cos(x)$ is even, but dividing by $x$ (odd) makes $\cos(x)/x$ an odd function.
(C) $\sin(x^2)$ is even because $(-x)^2 = x^2$, so $\sin(x^2)$ remains unchanged.
(D) $e^{-|x|}$ is even because $|-x| = |x|$.
Step 3: Conclusion.
Thus, the even functions are (C) and (D).