Identify Even Functions Analysis

A function $f(x)$ is defined as an even function if it satisfies the condition $f(-x) = f(x)$ for all values of $x$ in its domain.

Function Evaluation for Even Property

Let's evaluate each given function to determine if it is an even function:

• Option 1: $f(x) = |x|$

  To check if it's even, we evaluate $f(-x)$: $f(-x) = |-x|$ Since $|-x| = |x|$, we have $f(-x) = |x| = f(x)$. Therefore, $|x|$ is an even function.

• Option 2: $f(x) = \frac{\cos(x)}{x}$

  Evaluate $f(-x)$: $f(-x) = \frac{\cos(-x)}{-x}$ Using the property $\cos(-x) = \cos(x)$, we get: $f(-x) = \frac{\cos(x)}{-x} = -\frac{\cos(x)}{x}$ Since $f(-x) = -f(x)$, this function is odd, not even.

• Option 3: $f(x) = \sin(x^2)$

  Evaluate $f(-x)$: $f(-x) = \sin((-x)^2)$ Since $(-x)^2 = x^2$, we have: $f(-x) = \sin(x^2)$ Thus, $f(-x) = f(x)$. Therefore, $\sin(x^2)$ is an even function.

• Option 4: $f(x) = e^{-|x|}$

  Evaluate $f(-x)$: $f(-x) = e^{-|-x|}$ Since $|-x| = |x|$, we get: $f(-x) = e^{-|x|}$ Thus, $f(-x) = f(x)$. Therefore, $e^{-|x|}$ is an even function.

Conclusion on Even Functions

Based on the analysis, the functions that satisfy the condition $f(-x) = f(x)$ are $\sin(x^2)$ and $e^{-|x|}$.