Let f(x) = (2x+1)^3 and let g be the inverse function of f. Given that f(0)=1, what is the value of g'(1)?
In the inverse function, the ordered pairs are reversed.
So they're trying to get you to do this as easily as possible. Remember, what does an inverse function look like? It's just the same, except you've reflected it across the line y=x. This is great news, since that means the slope will also go through a similar transformation. So just take the derivative of f, and follow Mertsj's note to help you to the answer.
If (0,1) is part of the first function then (1,0) is part of the inverse function.
I replaced y with x and x with y. I get : x=(2y+1)^3. Now I don't know how to change it so that it's y=
Draw a picture and don't worry about the algebra right now. Don't waste your time solving it this way.
I don't understand :( How would I draw it?
Well you don't have to draw your exact function, just draw out a function like y=x^2 and it's inverse. See: https://www.google.com/search?q=inverse+function&espv=210&es_sm=93&source=lnms&tbm=isch&sa=X&ei=6bK3UrzXDerQyAHI-YGICw&ved=0CAkQ_AUoAQ&biw=1366&bih=704 There are a ton of examples there. See the relation between the graphs? Whatever the slope is on one, the slope on the inverse function is always the one over that slope. So if you have f(x)=mx+b then the slope on f^-1(x) is just going to be 1/m. look at the picture and make sure this is true for a couple simple graphs to understand why.
If G is the inverse of F, then that means that if F maps a point from point 1 to point 2, then G would map a point from point 2 to point 1.
Let f(x) be the function and g(x) be its inverse. If f(a) = b then g(b) = a Function f transforms a to b. The inverse function g transforms b back to a. The inverse function undoes the work of the function. If (a,b) is a point on f(x), then (b,a) will be the corresponding point on g(x). Relationship between the derivative of a function and the derivative of its inverse: Assume f(a) = b. Then, g(b) = a The slope of the inverse function at x = b is the reciprocal of the slope of the function at x = a (proof at the end) That is, g'(b) = 1 / f'(a) In this problem, a = 0, b = 1. Find g'(1). g'(1) = 1 / f'(0) f(x) = (2x+1)^3 f'(x) = 3(2x+1)^2 * 2 = 6(2x+1)^2 f'(0) = 6(0+1)^2 = 6 g'(1) = 1 / 6 Prove g'(x) = 1 / f'(g(x)) If g(x) is the inverse of f(x) then: g(f(x)) = x and f(g(x)) = x. Why? The function f transforms x to f(x). The inverse function transforms f(x) back to x. So g(f(x)) = x. The inverse function g undoes the work of the function f. The inverse function transforms x to g(x). The function f(x) transforms g(x) back to x. So f(g(x)) = x. The function f undoes the work of the inverse function g. Let us take f(g(x)) = x Take derivative on both sides. On the left use chain rule: f'(g(x)) * g'(x) = 1 g'(x) = 1 / f'(g(x))