Question 5 \[ f(x)=\frac{2 x}{x-1} \text { and } g(x)=x^{5}+1 \] Find \( f g(x) \), giving your answer in the form \( a x^{n}+b \) where \( a, b \), and \( n \) are integers.

To find \( f g(x) \), we begin by calculating \( g(x) \): \[ g(x) = x^5 + 1. \] Next, we can substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(x^5 + 1) = \frac{2(x^5 + 1)}{(x^5 + 1) - 1}. \] This simplifies the denominator: \[ (x^5 + 1) - 1 = x^5. \] Now, substituting this back into \( f(g(x)) \): \[ f(g(x)) = \frac{2(x^5 + 1)}{x^5}. \] Now, simplify the expression: \[ f(g(x)) = \frac{2x^5 + 2}{x^5} = 2 + \frac{2}{x^5}. \] To write this in the form \( ax^n + b \), we can express the term with an integer exponent: \[ \frac{2}{x^5} = 2x^{-5}. \] Therefore, combining terms gives us: \[ f g(x) = 2x^{-5} + 2. \] Here, \( a = 2 \), \( b = 2 \), and \( n = -5 \). So, the final answer is: \[ f g(x) = 2x^{-5} + 2. \]