Why is this (image) function transformation not logical?

Question

I am taking MIT 6.003 course from OCW and came upon the following problem from lecture 1:

For the above image $f(x, y)$, does the below image represent the transformation $f(-x-250, y)$?

The correct answer is no but my answer was yes and the reasoning was as follows:

First, we apply the image transformation using $f(-x, y)$ which will result in the original image being flipped about the vertical axis passing through x=0.

Next, we apply the image transformation using $f(x-250, y)$ which will result in the flipped image to translate to the right.

In total, intuitively speaking, applying $f(-x, y)$ followed by $f(x-250, y)$ should result in the 2nd image shown. Can anyone tell me what is the flaw in my logic? I can't figure out why this reasoning shouldn't work.

Answer 1

The photo should be toward $-250$ and not $250$ if this transformation is applied, it will be helpful to break this down into smaller transformations first simply consider $f(x+250,y)$ this is obviously towards $-250$ now when when we do $f(-(x+250),y)$ we get a flip and the image is still towards $-250$; Note that $f(-(x+250),y)=f(-x-250,y)$.

Answer 2

This is the case where translation may go first before scaling. To see why, let's recall that the function $f(x)$ translates to the positive direction when $a > 0$ and $f(x)$ becomes $f(x - a)$, and the function scales horizontally by a factor of $k$ when $f(x)$ becomes $f(kx)$. The special case is when $k = -1$ where it is a reflection.

What you did was reflect, then translate. By checking how that transformations apply, the plane reflects horizontally as expected. The translation part is where it breaks down. You are expecting the plane to be translated to the right (the positive direction), hence the $-250$, but you forgot that the plane was already reflected. This means that it will go to the left, not to the right.

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See this graph that I made from Desmos as an example. Note that they are points, and translating it to the positive direction will be to add, not to subtract.