Finding an Inverse Algebraically - Problem 1

Finding the inverse of an equation algebraically. So for this particular example we have a function that is a cubic and we want to find its inverse. The first thing we always have to do before finding the inverse is to make sure is make sure it's 1 to 1 so we actually have an inverse.

Going with what we know for graphs this is just a transformation of a cubic graph. We know that this graph is standard normally something like this and the minus 2 just moves it over to the right 2 units. Horizontal line test tells us the original is a function we move it over it's still going to be a 1 to 1 function so therefore we know that this is going to have an inverse. So then we can go ahead and solve that algebraically.

So first thing we need to remember is f(x) is actually the same thing as y. So replace that with y and we end up a y equals equation. To find the inverse switch x and y. So x is equal to y minus 2 quantity cubed and then we solve for y to get rid of the power we need to take the cube root. We end up with the cube root of x is equal to y minus 2, solving for y add 2 to the other side, and then lastly replace our y with a function inverse notation, f inverse of x is equal to root three write the cube root of x plus 2.

So finding the inverse of a function make sure it's 1 to 1, replace f(x) with y switch x’s and y’s then solve for y.