1
(1) Given f(x) = ---, evaluate and simplify the following:
(a) f(x) - f(5)
-----------
x - 5
(b) f(9 + h) - f(9)
---------------
h
(c) f(x + h) - f(x)
---------------
h
SOLUTION:
1 1
--- - ---
(a) f(x) - f(5) 
----------- = -----------
x - 5 x - 5
After substituting as above, rationalize the fractions in the numerator:
1 1
--- * --- - --- * ---
= -----------------------
x - 5
--- - ---
x 5
= -----------
x - 5
Next, get rid of the complex fraction:
5 x 5 x
--- * --- - --- * --- ----- - -----
x 5 5 x 5x 5x
= ----------------------- = ---------------
x - 5 x - 5
5 - x
-------------
5x 5 - x 5 - x
= ----------- = ----------- = -----------
x - 5 5x(x - 5) 5x2 - 25x
Then rationalize the numerator:
5 - x 5 + x
= ------------ * -----------
5x2 - 25x 5 + x
25x - 5x2
= -----------------------
(5x2 - 25x)(5 + x)
Finally, factor a -1 out of the numerator and reduce:
-(5x2 - 25x) -1
= ----------------------- = -------------
(5x2 - 25x)(5 + x) 5 + x
1 1 1 1
----- - --- ----- - ---
(b) f(9 + h) - f(9) 
3
--------------- = ------------- = -------------
h h h
1 3 1 3
----- * --- - --- * ----- ------- - -------
3 3 3 3
= ---------------------------- = -------------------
h h
3 -
--------------
3 3 -
= ---------------- = ------------
h 3h()
3 - 3 +
= ------------ * ------------
3h() 3 +
9 - (9 + h) 9 - 9 - h
= ------------------------ = ------------------------
3h()(3 + ) 3h()(3 + )
-h -1
= ------------------------ = ----------------------
3h()(3 + ) 3()(3 + )
1 1
------- - ---
(c) f(x + h) - f(x) 
--------------- = ---------------
h h
1 1
------- * --- - --- * -------
= -----------------------------
h
--------- - ---------
= ------------------------
h
-
--------------
-
= ---------------- = -------------
h h
- +
= ------------- * -------------
h +
x- (x + h) x - x - h
= ------------- = -------------
h h
-h -1
= ------------- = -------------
h h