If I have a function $f(x)$ and I then want to move it to the left by 2, I could represent this as $f(x + 2)$. Similarly, if I want to move the function to the right by 2, I could represent this as $f(x-2)$.
However, if I have $f(2-x)$ = $f(-x+2)$, I would think that this is a reflection across the y-axis and then a left shift of 2 (because the +2). However, it seems as though this is a reflection across the y-axis and then a right shift of two.
Why is this?
Thank you
Congratulations — you've (rediscovered) one of the simplest non-commutative groups. You're applying two operations: the 'shift by two' operator $S_2: x\mapsto x+2$ and the negate operator $N: x\mapsto -x$. But $N\circ S_2$ isn't the same as $S_2\circ N$; it depends on which order you do them in. We can see this by writing them out: $(N\circ S_2)(x) = -(x+2) = -x-2$, whereas $(S_2\circ N)(x)=2+(-x)=2-x$. Instead, we have $S_2\circ N = N\circ S_{-2}$; flipping the axis flips the sense of direction. (If you're curious, you can find more information about specific sets of operations like this under the name dihedral group; they're fundamental objects when we talk about symmetry.)