When a reflection over the y axis is applied first, does it apply to a horizontal translation?

Question

For example, if I reflect something over the y axis and it becomes $f(x) = -x$, does horizontally shifting it $2$ units right cause it to look like this: $$f(x) = -(x-2)?$$ Furthermore, if the horizontal shift is applied first, does the equation become $-(x-2)$ or $-x-2?$ My main question is, does the order of transformations matter when it is just horizontal translation, reflection in the y axis, or horizontal stretches.

Answer 1

A translation of a function horizontally two units to the right and then reflected about the y-axis:

Original Function: f(x)

Translated 2 to the right: f(x-2) = g(x)

Reflected about the y-axis: g(-x) = f(-x-2)

To understand more clearly, we can take a point on a function as an example. Let's use the function f(x) = x and take the point (3,3) on that function.

Beginning Point: (3,3)

Translated 2 units to the right: (5,3)

Reflected about y-axis: (-5,3)

Since our original function was f(x) = x, our new function should be f(-x-2) = -x-2. Since h(-5) = -(-5)-2 = 3, our function demonstrates these translations.

A function reflected about the y-axis and then shifted horizontally:

Original function: f(x)

Reflected about the y-axis: f(-x) = g(x)

Translated 2 units to the right: g(x-2) = f(-(x-2)) = f(-x+2)

If we take the point (3,3) again:

Reflected about y-axis: (-3,3)

Translated 2 units to the right: (-1,3)

Since our original function was f(x) = x, our new function should be f(-x+2) = -x+2.

Since h(-1) = -(-1) + 2 = 3, our function demonstrates these translations.

Although most problems arise when transforming functions horizontally, order does matter when transforming vertically as well. For example, an upward shift of a function 3, then a reflection about the x-axis: -(f(x) + 3) is much different than a reflection about the x-axis, then vertical shift 3: -f(x) + 3

Answer 2

I think of this as a composition of functions.

let $h(x) = x-2$ be our horizontal translation and $g(x) = -x$ be our reflection

$(f\circ g) (x) = f(-x)$ is our function reflected.

$(f\circ h) (x) = f(x-2)$ is our function translated.

$(f\circ g\circ h) (x)$ is our function first translated then reflected.

$(g\circ h) (x) = -(x-2) = 2-x\\ (f\circ g\circ h) (x) = f(2-x)$

$(f\circ h\circ g) (x)$ is our function first reflected then translated.

$(h\circ g) (x) = g(x) - 2 = -x -2\\ (f\circ g\circ h) (x) = f(-x-2)$