Evaluate the difference quotient for the given function.?

Question

f(x)=1/x, (f(x)-f(a))/x-a

Dont get it at all. Help please

Answer 1

f(x) = 1/x

f(a) = 1/a

f(x) - f(a) = 1/x - 1/a = (a - x) / (xa)

[f(x) - f(a)] / (x - a) = [(a - x) / (xa) ] / (x - a) = -1 / (xa)

as a gets closer to x, this becomes - 1 / x^2 (but that may be jumping ahead...)

Answer 2

If f(x) = 1/x then f(a) = 1/a

Let's plug in and see what happens

[(1/x) - (1/a)]/(x-a)

Find a common denominator for the numerator  (a*x  OR ax)

[(a/ax) - (x/ax)]/(x-a)

Combine over common denominator in numerator

[(a - x)/ax]/(x - a)

Which equals

[(a - x)/ax] * (1/(x - a))

Factor a -1 out of the numerator

[(-1)*(x - a)]/ax * [1/(x - a)]

Notice that the (x - a) cancels and all we are left with is

(-1)/ax

Answer 3

In referring to the frist answer to this question:

When you factored the (-1) out of the (a-x) in your second to last step, you forgot to change the sign to a positive, making it an (a+x), now one is able to cancel the (a+x) and the (a-x). Just a typo but it may clear up some confusion!

and as stated, you are left with -1/ax