BridgeFunctions and inverse functionsFree

Inverse functions and numerical solution (Newton's method)

mathematical

Let

f(x) = e^{2x} - 1

Facts you may need

$e^{2x} = u \iff x = \frac12\ln u$ — Undo the exponential.
Newton: $x_{n+1} = x_n - \frac{g(x_n)}{g'(x_n)}$ — Iterate until the digits settle.

Find

f^{-1}(x)

and state its domain and range.

Explain why any intersection of

y = f(x)

and

y = f^{-1}(x)

must lie on

y = x

, and show that the graphs meet at the origin.

Use Newton's method with

x_0 = -1

g(x) = e^{2x} - x - 1

to find the other intersection, correct to 4 decimal places.