Question

given the function: f(x)=x^2-2x. how can you restrict the domain so that f(x) has an inverse? what is the equation of the inverse function?

Answer 1

Solution

We have the following equation given:

$y=x^2-2x=x(x-2)$

For this case we can do the following:

$x=y^2-2y=y(y-2)$

We can solve for the quadratic equation and we got:

$y=\frac{-(-2)\pm\sqrt[]{(-2)^2-4(1\cdot-x)}}{2\cdot1}=\frac{2\pm\sqrt[]{4\cdot(1+x)}}{2}=1\pm\sqrt[]{1+x}$

The two solutions are:

$y_1=1-\sqrt[]{1+x},y_2=1+\sqrt[]{1+x}$

Then the answer is:

Part 1

$x\ge1$

Part 2

$f^{-1}(x)=1+\sqrt[]{1+x}$