Calculate g(0) if g is inverse to function f(x)=x^3+x-2?
The inverse function is the function obtained by exchanging x (input) values with y (output) values of the original function.
So, if the original function is `y = f(x) = x^3 + x - 2` , the inverse function g can be obtained by exchanging x and y:
`x = y^3 + y - 2` and then solving for y. Resultant expression would be y = g(x), the inverse function. In this case, this equation is difficult to solve. But it is not necessary since we only have to find
Set x = 0 and solve for y:
`y^3 + y - 2 = 0`
By inspection, we can find that y = 1 is one of the roots of this equation: `1^ 3 + 1 - 2 = 0`
The cubic polynomial above can be factored as
`y^3 -y^2 + y^2 - y + 2y - 2 = (y-1)(y^2 + y + 2) = 0`
The second factor is a quadratic trinomial that has no real roots.
So the value of the inverse function at x = 0 is `g(0) = 1` .