let F(x)=x-1/3x find f(a+h)-f(a)/h

sorry if this is difficult to understand.

f(x) = x - 1/3x ??

I think there are some missing brackets here, such as
f(x) = (x-1)/(3x) or some variation of that
or else why didn't they just say
f(x) = 2/3x ?

come back with the correction and I'll what I can do

yes sorry

let F(x)=(x-1)/(3x) find f(a+h)-f(a)/h.

f(x)=(x-1) over (3x) find f(a+h)-f(a) all over h.

ok then

f(a) = (a-1)/(3a)
f(a+h) = (a+h - 1)/(3(a+h))

(f(a+h)-f(a) )/h
= ( (a+h - 1)/(3(a+h)) - (a-1)/(3a) )/h
= ( a(a+h - 1) - (a+h)(a-1) )/(3a(a+h)) * 1/h
= (a^2 + ah - a - a^2 + a - ah + h)/(3a(a+h)) * 1/h
= h/(3a(a+h)) * 1/h
= 1/(3a(a+h))

I am confused by the multiplication of 1/h.

No problem! I can help you with that.

To find the value of [f(a+h) - f(a)] / h, we need to substitute the expressions for f(a+h) and f(a), and then simplify the expression.

First, let's find f(a+h):

f(a+h) = (a+h) - (1/3)(a+h)

Next, let's find f(a):

f(a) = a - (1/3)a

Now, let's substitute these expressions back into [f(a+h) - f(a)] / h:

[f(a+h) - f(a)] / h = [(a+h) - (1/3)(a+h) - (a - (1/3)a)] / h

Now, simplify the expression by combining like terms:

= [(a+h) - (1/3a + 1/3h) - a + (1/3)a] / h

= [(a - a) + h - 1/3a - 1/3h + 1/3a] / h

= [h - 1/3h] / h

= [h(1 - 1/3)] / h

= [h(2/3)] / h

= 2/3

Therefore, the value of [f(a+h) - f(a)] / h is 2/3.