Calculate the domain and range for this equation $y^{2}=-2(x-20)\{x> 10.4\}$

Step 1 ：First, we need to find the domain of the function. Since the function is defined for \(x > 10.4\), the domain is \((10.4, \infty)\).

Step 2 ：Next, we find the range of the function. We have \(y^2 = -2(x - 20)\), so \(y^2 = -2x + 40\).

Step 3 ：Since \(y^2\) is always nonnegative, we must have \(-2x + 40 \ge 0\), which gives us \(x \le 20\).

Step 4 ：Combining this with the domain, we have \(10.4 < x \le 20\).

Step 5 ：Now, we can rewrite the equation as \(y^2 = 40 - 2x\).

Step 6 ：Since \(y^2\) is always nonnegative, the minimum value of \(y^2\) is 0, which occurs when \(x = 20\). Thus, the minimum value of \(y\) is 0.

Step 7 ：The maximum value of \(y^2\) occurs when \(x = 10.4\), which gives us \(y^2 = 40 - 2(10.4) = 19.2\). Thus, the maximum value of \(y\) is \(\sqrt{19.2}\).

Step 8 ：Since \(y^2\) can take on any value between 0 and 19.2, the range of \(y\) is \([-\sqrt{19.2}, \sqrt{19.2}]\).

Step 9 ：Thus, the domain is \((10.4, 20]\) and the range is \(\boxed{[-\sqrt{19.2}, \sqrt{19.2}]}\).