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?

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

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?

Thank you very much

To simplify the expression [f(5+h)−f(5)]/h, let's start by substituting the values you calculated correctly: f(5+h) = 1/(14+h) and f(5) = 1/14.

Now, let's subtract these two expressions:

[f(5+h)−f(5)] = [1/(14+h) - 1/14]

To find a common denominator between 14+h and 14, we multiply the denominator and numerator of the first term by 14:

[f(5+h)−f(5)] = [1/(14+h) - (14/14h)]

Simplifying the first term:

[f(5+h)−f(5)] = [1/(14+h) - (14/14h)] = [1/(14+h) - (14/(14h)) * (h/h)]

Combining the terms:

[f(5+h)−f(5)] = [(1 - 14h/(14(14+h)))] / h

Simplifying the numerator further:

[f(5+h)−f(5)] = [(1 - h/(14+h)) / h]

Now, to match the desired format −1/(ah+b), let's multiply the numerator and denominator by (-1):

[f(5+h)−f(5)] = [-1 * (h/(14+h) - 1) / -h]

Which simplifies to:

[f(5+h)−f(5)] = [(1 - h/(14+h)) / h] * (-1)

[f(5+h)−f(5)] = [h/(h + 14) - 1] / h

Now, we can see that a = 1 and b = 14 in the format −1/(ah+b). Therefore, the final expression is:

[f(5+h)−f(5)] = [h/(h + 14) - 1] / h = -1/(1*h + 14)

So, a = 1 and b = 14.

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!

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