Evaluate the function at the given value of the independent variable?

Question

Show me how u got it. Evaluate the function at the given value of the independent variable. Simplify the results.

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

Find:

[f(x) - f(2)] / (x-2)

The answer in the back of the book is:

-1/{sqrt(x-1)[1+sqrt(x-1)]}, where x does not equal 2

Answer 1

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

Evaluate the function and substitute:

(1/√(x-1) - 1/√(2-1)) / (x-2)

Simplify:

(1/√(x-1) - 1) / (x-2)

Extract a factor of 1/√(x-1):

1/√(x-1) (1-√(x-1))/(x-2)

Rationalize the numerator by multiplying both numerator and denominator by 1+√(x-1):

1/(√(x-1) (1+√(x-1)) (1-(x-1))/(x-2)

Simplify:

1/(√(x-1) (1+√(x-1)) (2-x)/(x-2)
-1/(√(x-1) (1+√(x-1))

Which is the answer in the back of your book. The restriction x≠2 is necessary, because in the final form this expression can be evaluated at 2, whereas the original expression could not, so this is needed to remind you that this is only equivalent to the original expression when x is not actually equal to 2.