Step 1 :The function is a rational function because it has vertical and horizontal asymptotes. The x-intercepts are the roots of the numerator and the vertical asymptote is the root of the denominator. The y-intercept is the value of the function at x=0. The horizontal asymptote is the limit of the function as x approaches infinity or negative infinity.
Step 2 :Consider the following possible equations for the function: \(f(x)=\frac{(x-4)(x+2)}{(x-1)^{2}}\), \(f(x)=\frac{(x-4)}{(x-1)^{2}}\), \(f(x)=\frac{(x-1)^{2}}{(x-4)(x+2)}\), \(f(x)=\frac{(x-4)(x+2)}{(x-1)}\).
Step 3 :By analyzing the results, we can see that the first function \(f(x)=\frac{(x-4)(x+2)}{(x-1)^{2}}\) has the correct x-intercepts (0,0), y-intercept (-8), vertical asymptote (-oo), and horizontal asymptote (1,1).
Step 4 :\(\boxed{f(x)=\frac{(x-4)(x+2)}{(x-1)^{2}}}\) is the correct equation for the function with the given features.