Let f(x) be the function 1/(x+9). Then the quotient [f(5+h)−f(5)]/h can be simplified to −1/(ah+b). What does a and b equal to?

Question

So I understand the overall concept, and did f(5+h) = 1/(14+h) and f(5) = 1/14.
Then to subtract the two, you get the common denominator which is 14(14+h).
So when you subtract it becomes (14-14+h)/[14(14+h)].
And then you divide that by h and get (14h-14h+h^2)/(196+14h), which is simplified to h^2/(196+14h).

My question is how do you get this into the format −1/(ah+b) if h^2 is in the numerator right now?

please explain and thanks

Answer 1

why did you multiply the numerator by h when you were supposed to divide by h?

dividing (14-14+h)/[14(14+h)] by h gives

h/[14h(14+h)] = 1/(196+14h)

You were so close!

Answer 2

Yeah, that was a stupid mistake...

But now I have another question. Isn't the final answer supposed to be in −1/(ah+b) form? Why is my answer 1/(196+14h) still positive?

Did I do something else wrong?

Answer 3

1/(14+h) - 1/14 = (14-(14+h))/[14(14+h)] = -h/[(14(14+h)]

should have caught that, eh?
1/(14+h) < 1/14

so the result will be negative

next time don't forget the parentheses

Answer 4

Thank you very much