Odd and even functions ：練習問題

Which of the following functions satisfies the property that its graph is symmetric with respect to the \(y\)-axis?

Let \(f(x) = \frac{2^x + 1}{2^x - 1} \sin|x|\) for all non-zero real numbers \(x\). Which of the following statements about the function \(f(x)\) is true?

Suppose that the function \( f(x) = \frac{x}{3^x - 1} + \frac{x}{k} \) is an even function for all non-zero real numbers \( x \), where \( k \) is a non-zero constant. Find the value of \( k \).

Let \(f(x)\) be an odd function and \(g(x)\) be an even function, both defined for all real numbers. Determine the parity of the function \(h(x) = [f(x)]^2 + g(x)\).

Let \(f(x)\) be a non-zero even function and \(g(x)\) be a non-zero odd function defined for all real numbers. Consider the function \(h(x) = \frac{f(x) \cdot [g(x)]^2}{x \sin x}\) for \(x \neq 0\). Which of the following statements about \(h(x)\) is true?

Determine whether the function \(f(x) = x^2 \cos x + |x|\) is an odd function, an even function, or neither.

A function \(h(x)\) is defined by \(h(x) = \frac{e^{x^2} \sin x}{x^4 + 1}\). By considering the symmetry of its components, determine whether \(h(x)\) is an odd function, an even function, or neither.

A differentiable function \(f(x)\) defined on \(\mathbb{R}\) is expressed as the sum of an even function \(g(x)\) and an odd function \(h(x)\). If \(g(x) = x^4 + 3\) and the derivative \(h'(x)\) satisfies \(h'(x) = e^x + e^{-x}\), determine the value of \(f(\ln 2)\).