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Expert: - 8/30/2008


Question
Evaluate the difference quotient for the given function. Simplify your answer.

F(x) = x + 3 /x + 1                 , F(x) - F(1)/ x - 1

Answer
Take F(x) from the first equation and put it into the second equation.
This gives you x+3/x+1 - (x+3/x+1)/x - 1.

The parenthesis are there due to the fact that the entire function is divided by x.

Should F(x) really have parenthesis?  I looked at the problem for a few minutes, and realized that F(x) should really be
F(x)=(x+3)/(x+1).  I figure the parenthesis were not included since
F(x) was originally in two lines and didn't need to have them.

In that case, to evaluate (F(x)-F(1))/(x-1), we need to know the value of F(x) at x=1.  This is (1+3)/(1+1)=2.

Now that this is known, the function is
((x+3)/(x+1) - 2)/(x-1).

To combine the two termns on top, we must first give the 2 the same denominator as the (x+3)/(x+1) by multilplying the 2 by (x+1)/(x+1), giving us [(x+3 - 2(x+1))/(x+1)]/(x-1).

This reduces to [(x+3-2x-2)/(x+1)]/[(x-1)/1].  In the bottom, the /1 was added since we must be dividing two fractions.  Remember if we have a complex fraction (a/b)/(c/d), the result is (ad)/(bc) since the we invert the bottom and multiply.  In this case we get
(-x+1)/((x+1)(x-1)) = -(x-1)/[(x+1)(x-1)].

Here, the x-1 cancel giving us -1/(x+1).  Note that here, since we divided the top and bottom by x-1, we have x can't equal 1, but we can get as close to 1 as we want.

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