Most of the (basic) math I've been doing in school has been easily checkable - it is immediately obvious if one had the right answer by just plugging it back in. Recently, doing function transformations, we were not taught a method to check if a transformed function was correct, and I am unable to figure it out myself.
Say we had an original function - $x^2-1%$ for example. I wanted to translate this function 2 units right, then reflect horizontally (over the y-axis). So I came up with the equation $(-x-2)^2-1$.
Plotting these two functions, with the original on the right:
http://i.stack.imgur.com/TKjiy.png (Image of the two graphs)
I can start at a point like $(0, -1)$, then add 2 units right, and reflect, and easily see that this transformation is correct. However, I would like to check algebrically if possible.
I've tried inputting the x coordinate in the transformed function and then trying to solve to see if I get the same y value, to no avail. I'm not sure how I can even guarantee a point exists on the original graph!
Summary: Given an original function, and a transformed function, is it possible to check if the transformed function satisfies applied transformations?