1. **Problem:** Determine which statement is true for the power function $f(x) = x^n$ where $n$ is a negative even integer. 2. **Recall:** A power function $f(x) = x^n$ with $n$ negative even integer means $n = -2, -4, -6, \dots$. 3. **Key properties:** - Since $n$ is even, $f(x)$ is symmetric about the y-axis. - Since $n$ is negative, $f(x)$ involves reciprocal powers, so $f(x) = x^n = \frac{1}{x^{|n|}}$. - The function is undefined at $x=0$ (division by zero). - As $x \to \pm \infty$, $f(x) \to 0$, so there is a horizontal asymptote at $y=0$. - The function is always positive for all $x \neq 0$ because even powers are positive. 4. **Check each option:** - A) "The function is always increasing for all $x$ in its domain." This is false because the function decreases on $(0, \infty)$ and also on $(-\infty, 0)$. - B) "The function is symmetric about the origin and passes through $(1,1)$." Symmetry about the origin means odd function, but even powers are symmetric about y-axis, so false. - C) "The range of the function is all real numbers." Since $f(x) > 0$ for all $x \neq 0$, range is $(0, \infty)$, so false. - D) "The function is symmetric about the y-axis and has a horizontal asymptote at $y=0$." This is true. **Final answer:** D