Find f(x+h) -f(x) all over h...given f(x)= sqrt of all over 2x +3 . Rationalize the numerator if possible.
it doesnt say what f(x) is the sqrt of?
If you use parentheses, you would be able to see the expression and give it a good check also. For example
(f(x+h)-f(x))/h where f(x) = sqrt(2x+3)
Be very careful when you interpret fractions. Even the expression does not show parentheses, you need to insert parentheses around the numerator and the denominator, like
(x-4)/(x²-16) and not x-4/x²-16.
Back to the origina question:
(f(x+h)-f(x))/h where f(x) = sqrt(2x+3)
The simple answer would be:
(sqrt(2(x+h)+3)-sqrt(2x+3))/h
=(sqrt((2(x+h)+3)/h²)-sqrt((2x+3)/h²))
This is about as far as you can go.
If you are eventually taking the limit h->0, then you can further simplify using series expansions.
Actually, you can rationalize the numerator as follows by multiplying both the numerator and denominator by: (sqrt(2(x+h)+3)+sqrt(2x+3))
(sqrt(2(x+h)+3)-sqrt(2x+3))/h
=(2(x+h)+3)-(2x+3)/(h(sqrt(2(x+h)+3)+sqrt(2x+3)))
=(2h)/(h(sqrt(2(x+h)+3)+sqrt(2x+3)))
=2/(sqrt(2(x+h)+3)+sqrt(2x+3))
To find f(x+h) - f(x) all over h, we need to substitute f(x+h) and f(x) into the equation.
First, let's find f(x+h):
f(x+h) = sqrt(2(x+h) + 3) / 2
Next, let's find f(x):
f(x) = sqrt(2x + 3) / 2
Now we can substitute these expressions into the equation:
f(x+h) - f(x) = [sqrt(2(x+h) + 3) / 2] - [sqrt(2x + 3) / 2]
To rationalize the numerator, we can multiply both the numerator and denominator by the conjugate of the numerator, which is sqrt(2(x+h) + 3) + sqrt(2x + 3):
[f(x+h) - f(x)] / h = [sqrt(2(x+h) + 3) / 2 - sqrt(2x + 3) / 2] * [(sqrt(2(x+h) + 3) + sqrt(2x + 3)) / (sqrt(2(x+h) + 3) + sqrt(2x + 3))]
Simplifying this expression, we get:
[f(x+h) - f(x)] / h = [sqrt(2(x+h) + 3)(sqrt(2(x+h) + 3) + sqrt(2x + 3)) - sqrt(2x + 3)(sqrt(2(x+h) + 3)))] / [2(sqrt(2(x+h) + 3))(sqrt(2x + 3))]
Now you can simplify and further manipulate the expression if needed.