Given f(x)=x+3/x+2, simplify f(x+h)-f(x)/h, when x=4

Question

(4+h+(3/4+h)+2)-(4+3/4+2)/h

(4+h+(3/4+h)+2-4-3/4-2)/h

(h+(3/4+h)-3/4)/h

h+(3/4+h)-(3+h/4+h)/h

and there is where I am stuck?

Answer 1

f(x) = (x+3)/(x+2)

 

f(4+h) - f(4) = (7+h)/(6+h) - 7/6 = [6(7+h) - 7(6+h)]/[6(6+h)]

 

                                                              = -h/[6(6+h)]

 

So, [f(4+h) - f(4)]/h = -h/[6(6+h)] ÷ h

 

                                       = -h/[6(6+h)] · (1/h) 

 

                                       = -1/[6(6+h)]

Answer 2

With problems like this one, it is really important to use parentheses so that P-E-M-D-A-S keeps the operations in order.

Now, the confusing part is that f(x) is given as a pattern equation.  Remember that.

The problem is to find f(z+h)-f(z)/h when z=4      [PLZ allow me to un-confuse this]

 

So, here we go, find:

     f(4+h) – f(4)/h

       (4+h) + 3/(4+h) + 2    -   [ 4 +3/4 + 2 ] / h           (write the difference of functions)

       6 + h + 3/(4+h)      -   27/(4h)                  (just collecting terms)

       h + 3/(h+4) – 27/(4h) + 6                 (often written in descending powers of h