Use the given functions to find, simplify, and identify the domain of the following combinations \[ f(x)=\frac{4 x+7}{x-4} \text { and } g(x)=\frac{-1 x+9}{x-2} \] \[ (f+g)(x)= \] Domain of $(f+g)(x)$ : \[ (f-g)(x)= \] Domain of $(f-g)(x)$ : \[ (f g)(x)= \] Domain of $(f g)(x)$ :

Step 1 ：\((f+g)(x) = f(x) + g(x) = \frac{4 x+7}{x-4} + \frac{-1 x+9}{x-2}\)

Step 2 ：To add these two fractions, we need to find a common denominator. The common denominator of \(x-4\) and \(x-2\) is \((x-4)(x-2)\)

Step 3 ：\((f+g)(x) = \frac{(4 x+7)(x-2) + (-1 x+9)(x-4)}{(x-4)(x-2)}\)

Step 4 ：Simplify the numerator: \(= \frac{4x^2 - 8x + 7x - 14 - x^2 + 4x - 36}{(x-4)(x-2)} = \frac{3x^2 + 3x - 50}{(x-4)(x-2)}\)

Step 5 ：\(\boxed{(f+g)(x) = \frac{3x^2 + 3x - 50}{(x-4)(x-2)}}\)

Step 6 ：\((f-g)(x) = f(x) - g(x) = \frac{4 x+7}{x-4} - \frac{-1 x+9}{x-2}\)

Step 7 ：\((f-g)(x) = \frac{(4 x+7)(x-2) - (-1 x+9)(x-4)}{(x-4)(x-2)}\)

Step 8 ：Simplify the numerator: \(= \frac{4x^2 - 8x + 7x - 14 + x^2 - 4x - 36}{(x-4)(x-2)} = \frac{5x^2 - 5x - 50}{(x-4)(x-2)}\)

Step 9 ：\(\boxed{(f-g)(x) = \frac{5x^2 - 5x - 50}{(x-4)(x-2)}}\)

Step 10 ：\((fg)(x) = f(x) \cdot g(x) = \frac{4 x+7}{x-4} \cdot \frac{-1 x+9}{x-2}\)

Step 11 ：\((fg)(x) = \frac{(4 x+7)(-1 x+9)}{(x-4)(x-2)}\)

Step 12 ：Simplify: \(= \frac{-4x^2 + 36x - 7x + 63}{(x-4)(x-2)} = \frac{-4x^2 + 29x + 63}{(x-4)(x-2)}\)

Step 13 ：\(\boxed{(fg)(x) = \frac{-4x^2 + 29x + 63}{(x-4)(x-2)}}\)