Algebraic Combinations. Consider the function

Question

f(x) = x+1/x-1

Calculate the difference quotient

f(x)-f(2)/x-2

Answer 1

There is a question about "Calculate the difference quotient".

In Calculus, the difference quotient is (f(x+h)-f(x))/h. But I don't think that is what is being asked here.

I believe the question is to simplify:

(f(x)-f(2))/(x-2) where f(x) = (x+1)/(x-1)

f(2) = (2+1)/(2-1) = 3

(f(x)-f(2))/(x-2) =

((x+1)/(x-1) -3)/(x-2) = (x+1-3x+3)/((x-1)(x-2)) = (-2x+4)/((x-1)(x-2)) = -2(x-2)/((x-1)(x-2)) = -2/(x-1)

Answer 2

You forgot to put parenthesis. The above solution is completely correct so:

f(X) - f(2) = (X + 1) / (X - 1) - 3 = ( -2X + 4 ) / (X - 1) using common denominator.

After dividing by (X - 2) we will get:

f(X) - f(2) / (X -2) = -2(X - 2) / ((X - 1)(X - 2)) = -2 / (X - 1).