1. The problem asks which functions can be represented by a Fourier series over the given ranges. 2. A Fourier series represents a function as a sum of sines and cosines over a finite interval, typically periodic or extended periodically. 3. Important rules: - The function must be periodic or defined on a finite interval and extended periodically. - It must be piecewise continuous and have a finite number of discontinuities. - Functions defined on infinite intervals without periodicity cannot be represented by a Fourier series. 4. Analyze each function: (a) $\tanh^{-1}(x)$ on $(-\infty, \infty)$: This function is defined on the entire real line, not periodic, so it cannot be represented by a Fourier series. (b) $\tan x$ on $(-\infty, \infty)$: $\tan x$ is periodic with period $\pi$, but it has vertical asymptotes (discontinuities) at $x=\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots$. Since it has infinite discontinuities in any interval of length $2\pi$, it is not piecewise continuous on a finite interval and thus not suitable for Fourier series representation over $(-\infty, \infty)$. (c) $|\sin x|^{-1/2}$ on $(-\infty, \infty)$: This function is periodic with period $2\pi$. However, $|\sin x|^{-1/2}$ has singularities where $\sin x=0$, i.e., at multiples of $\pi$, where it tends to infinity. Since it is not piecewise continuous (infinite discontinuities), it cannot be represented by a Fourier series. (d) $\cos^{-1}(\sin^2 x)$ on $(-\infty, \infty)$: This function is periodic because $\sin^2 x$ is periodic with period $\pi$, and $\cos^{-1}$ is continuous on $[0,1]$. The function is bounded and piecewise continuous on any finite interval. Therefore, it can be represented by a Fourier series. (e) $x \sin(1/x)$ on $(-\pi^{-1}, \pi^{-1}]$, cyclically repeated: The function is defined on a finite interval and extended periodically. It is bounded and piecewise continuous (the limit at 0 is 0). Hence, it can be represented by a Fourier series. 5. Final answer: - Functions (d) and (e) can be represented by Fourier series over the indicated ranges. \boxed{\text{(d) and (e) only}}