Given the function $y=cot\left(x\right)$, which of the following statements is true about its inverse function $y=arccot\left(x\right)$?

• A. The graph of $arccot\left(x\right)$ is symmetric about the line $y=x$.
• B. The inverse function $arccot\left(x\right)$ is defined for $x>0$ only.
• C. The range of $arccot\left(x\right)$ is $(0,\pi )$.
• D. The domain of $arccot\left(x\right)$ is all real numbers.

Answer: (D)

Verified step by step guidance

1. Recall that the function \( y = \cot(x) \) is the cotangent function, which is the reciprocal of the tangent function, defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
2. Understand that the inverse function \( y = \arccot(x) \) reverses the roles of \( x \) and \( y \), meaning \( \arccot(x) \) gives the angle whose cotangent is \( x \).
3. Identify the domain and range of the original function \( \cot(x) \) on its principal interval. The cotangent function is typically considered on the interval \( (0, \pi) \) where it is continuous and strictly decreasing, which ensures the inverse exists.
4. From the principal interval of \( \cot(x) \), deduce that the range of \( \arccot(x) \) is \( (0, \pi) \), because the inverse function's range corresponds to the original function's domain.
5. Note that the domain of \( \arccot(x) \) is all real numbers \( (-\infty, \infty) \) since cotangent can take any real value, and the graph of \( \arccot(x) \) is symmetric about the line \( y = x \) because it is the inverse of \( \cot(x) \).