Let f be the increasing function defined by f(x)=x^3+2x^2+4x+5, where f(-1)=2. If g is the inverse function of f, what is the tangent line equation to g at x=2?

Answer 1

Before we begin, let's pull out the important points of information.

1. f(x) = x³ + 2x² + 4x + 5
2. f(-1) = 2
3. g is the inverse function of f

Why is #3 important? What information does it give us? Well, inverse means that f(g(x)) = x and g(f(x)) = x. In this context that means that if I plug -1 into f(x) and get 2, I'll get -1 if I plug 2 into g(x).

That also means the slope at point (-1, 2) on the graph f(x) multiplied by the slope at point (2, -1) on the graph g(x) will be 1. So ∂f/∂x(-1) * ∂g/∂x(2) = 1. Rewritten, we have ∂g/∂x(2) = 1/(∂f/∂x(-1)). Let's find ∂f/∂x(-1).

∂f/∂x = 3x² + 4x + 4

∂f/∂x(-1) = 3

So ∂g/∂x(2) = 1/3

Now we have the slope and a point on the line. Let's solve for the line.

y + 1 = 1/3 (x - 2)

y = 1/3x - 5/3