Question involving inverse of a cubic function

If g(x)= x³ + 3x² -12x, where x belongs to the set of real numbers from -3 to 1 (both values inclusive), evaluate g^-1 (11).

I don' know how to find the inverse of a cubic equation……..

Student X

You are not required to find the actual expression for its inverse in this particular case.

This is how the inverse function really works: for a point (x,y) on the original function, this point is mapped to (y,x) on the inverse function such that its x and y coordinates are merely being swapped. So, a point (1,2) on the original function implies a point (2,1) will reside on the inverse function. Hence the so called y=x mirror line you were taught to draw.

So, for this particular cubic function,11 is the x coordinate of g^-1 (where you are required to find its y value of g^-1 (11) ) and is actually also the y coordinate of g, so we can set g(x)= 11, ie  x^3 + 3x^2 -12x =11. Solving this using the graphic calculator gives 3 possible values of x:  x= -4.97,  x=2.77 or x= -0.8 .

However, note that the range of g^-1 (x) equals the domain of g(x), hence the only accepted value is -0.8 ( between -3 and 1). As such we have g^-1 (11)=  -0.8.

Hope this helps. Peace.

Best Regards,

Mr Koh