Is there a particular order to doing reflections when there is an inverse reflection (about the line y=x) involved?

Question

For example, the original function would be y = f(x) and the transformed would be x = -f(y)

If an x-intercept on the original graph was (-2,0) depending on which function was done first, the resulting y-intercept on the reflected graph can be (0,-2) or (0,2).

Also, with the graph of x = -f(y) would the reflection be around the x-axis? or the y-axis?

I had a few other questions where it seemed like for inverse graphs an x = -f(y) would indicate a reflection about the y-axis and x = f(-y) would indicate a reflection about the x-axis

Answer 1

For any curve $y=f(x)$, by reflection, we can make the following transformations.

Let this be the graph of $y=f(x)$

Transformations

• $x\to {-x}$ is reflection about y-axis and gives us the graph of $y=f(-x)$

• $y\to -y$ is reflection about x-axis and gives us the graph of $y=-f(x)$

• $x\to y$ and $y\to x$ is reflection about $x=y$ line and gives us the graph of $x=f(y)$

• $x\to -y$ and $y\to -x$ is reflection about $x=-y$ line and gives us the graph of $-x=f(-y)$

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Now, moving on to your question, to transform $y=f(x)\to x=-f(y)$, We could do the following changes $$y=f(x)\to x=f(y)$$ $$x=f(y)\to -x=f(y)\implies x=-f(y)$$ So, first take reflection about $y=x$ then refelect it about y-axis.

I hope this lefts no doubts remaining. Cheers!