help with inverse function?

For part B, the easy way is to see that f(1) = 7. 1+2+4 = 7. So, f^{-1}(7) = 1. The hard way would be to try to determine the formula for the inverse, but let's not do that.

As for the derivative, it's helpful to realize that the graph of the inverse function is always a mirror image (over the line x=y) of the graph of the original function. That is, if you take your graph of f, but think of the horizontal as the y axis, and the vertical as the x axis, you get the graph of g. So g'(7) = 1/(f'(g^{-1}(7)) = 1/f'(1).

Notice that f(1) = 7. To the extent that f(x) admits an inverse (which this function does), then fֿ¹(7) = 1

Answer (b): 1

f'(x) = 5x^4 + 6x^2 + 4

f'(1) = 15 .............................. you'll need this later

f(fֿ¹(x)) = x ............................. an inverse by definition

f'(fֿ¹(x)) fֿ¹'(x) = 1 .................... chain rule

fֿ¹'(x) = 1 / f'(fֿ¹(x))

fֿ¹'(7) = 1 / f'(fֿ¹(7)) = 1/f'(1) = 1/15

Answer (c): fֿ¹'(7) = 1/15

Part B is just plain silly. In part A it says g = f^-1, so g(7)=f^-1(7) MUST be true!

Part C, plug 7 in for the variable in g and then simplify it.