Explain how the graph of the function $f(x)=\frac{-1}{(x+4)^{2}}-2$ can be obtained from the graph of $y=\frac{1}{x^{2}}$. Then graph $f$ and give the (a) domain and (b) range. Determine the largest open intervals of the domain over which the function is (c) increasing or (d) decreasing. To obtain the graph of $\mathrm{f}$, shift the graph of $y=\frac{1}{x^{2}} \square \nabla 4$ units, reflect across the $\square$ and shift 2 units

Step 1 ：The function $f(x)=\frac{-1}{(x+4)^{2}}-2$ can be obtained from the graph of $y=\frac{1}{x^{2}}$ by shifting the graph 4 units to the left, reflecting it across the x-axis, and shifting it 2 units down.

Step 2 ：The domain of the function is all real numbers except -4, because the function is undefined at x = -4. So, the domain is $(-\infty, -4) \cup (-4, \infty)$.

Step 3 ：The range of the function is all real numbers less than -2, because the function is always less than -2. So, the range is $(-\infty, -2)$.

Step 4 ：The function is increasing or decreasing in the intervals where its derivative is positive or negative, respectively. The derivative of the function is $\frac{2}{(x + 4)^{3}}$.

Step 5 ：The function is increasing for $x > -4$ and decreasing for $x < -4$. So, the function is increasing on the interval $(-4, \infty)$ and decreasing on the interval $(-\infty, -4)$.

Step 6 ：Final Answer: (a) The domain of the function is $\boxed{(-\infty, -4) \cup (-4, \infty)}$. (b) The range of the function is $\boxed{(-\infty, -2)}$. (c) The function is increasing on the interval $\boxed{(-4, \infty)}$. (d) The function is decreasing on the interval $\boxed{(-\infty, -4)}$.