Finding Symmetry of Functions
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A function is even if it is symmetric with respect to reflection about the \( y \)-axis. and odd if it is symmetric with respect to \( 180^\circ \) rotation.
Any even function \( f \) must satisfy \( f(x) = f(-x). \)
Any odd function \( f \) must satisfy \( f(x) = -f(-x). \)
The function specified by \( f(x) = x^2 \) is even while \( f(x) = x^3 \) is odd.
A function can also display periodic symmetry. A function that repeats infinitely for a given fixed distance along the \( x \)-axis is said to be a periodic function, with the fixed distance called the period.
A function \( f \) is periodic if
\[ f(x) = f(x + T) \]
for all \( x \) for some nonzero \( T \). The smallest positive \( T \) for which \( f \) is periodic is the primitive period of the function.
The word "period" alone is often used to refer to the primitive period.
Because the sine function repeats every \( 2\pi \) radians \(\big(\)that is, \( \sin{x} = \sin(x + 2\pi)\big) \), it has a period of \( 2\pi \).
We all know that by parity of integers,
\[ \text{ odd number + odd number = even number } \; . \]
Then \(\text{ odd function + odd function }\) is always an \(\text{__________} \).
Clarification: The function in question is for \(\mathbb R \to \mathbb R\).
Determine whether the function \(f(x) = \ln(x+ \sqrt{1+x^2} )\) is an odd function or an even function.