Expert: Paul Klarreich - 1/29/2008

Question
Hi, can you help me with finding the difference quotient for the function, f(x)= x/x-2?  Thanks a bunch.

Answer
Questioner:   Kenton
Category:  Advanced Math
Private:  No
 
Subject:  Difference Quotient
Question:  Hi, can you help me with finding the difference quotient for the function, f(x)= x/x-2?  Thanks a bunch.
................................
Hi, Kenton,

The D.Q. is

f(x + h) - f(x)
----------------
     h

which means exactly what it says:

1. Write the result of substituting (carefully)  x+h for x in the function.
2. Write the result of substituting (carefully)  x for x in the function, which should not be hard to do.
3. Subtract. [That's how you get a difference.]
4. Divide by h. [That's how you get a quotient.]
5,6,7,... Simplify (demonstrate that you didn't sleep through your elementary algebra class)

To write f(x + h):

--- write f(x), putting empty parentheses in place of x:  [I mean it!]
        ( )
f( ) = -------
      ( ) - 2

Put the stuff into the parentheses:
            (x + h)
f(x + h) = ------------
          (x + h) - 2
maybe simplify a bit:

            x + h
f(x + h) = ------------
          x + h - 2

Now f(x), the same way:

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

Subtract:
                   x + h         x
f(x + h) - f(x) = ---------- - -------
                 x + h - 2     x - 2  

Do some algebra, like combining.  Do it right -- the LCD is (x + h - 2)(x - 2)

                 (x + h)(x - 2) - x(x + h - 2)
f(x + h) - f(x) = -----------------------------
                     (x + h - 2)(x - 2)

Simplify a bit:

                 x^2 + hx - 2x - 2h - x^2 - hx + 2x
f(x + h) - f(x) = -----------------------------------
                     (x + h - 2)(x - 2)

                     - 2h  
f(x + h) - f(x) = --------------------
                  (x + h - 2)(x - 2)

NOW you can do the dividing by h:  The term on top gets divided by h:

                       - 2  
f(x + h) - f(x) = --------------------
                  (x + h - 2)(x - 2)

and you are done.