Graph of Function With Translation

Question

I tutor a student and the question they have is: " Give an example of a function whose horizontal translation gives the same resulting graph as the vertical translation ". My work: I thought of several well known collegiate functions like linear, quadratic, exponential and they don't work. What about $f(x) = x$ ? For example, the horizontal shift right by $1$ unit gives the same graph as vertical shift down by $1$ unit. I wonder if there can be other functions? I haven't spent enough time on this but preliminary that's what I got. Any input is appreciated. Thanks, WY.