Question

Can `y=6` be considered a function of `x`?

Answer 1

Yes it can.

Explanation:

A function is a mapping from one set (of numbers) to another. As long as each input gets mapped to only one output (e.g. the mapping doesn't take the input "3" and map it to both "9" and "27"), the mapping can be considered a function.

A function is like a computer program. It's been programmed to do the same thing to every input it gets. We give it an input, the function takes that input, does something to it, and returns an output. If we give it that same input again, we should get the same output again.

For example, a function `y` may be defined as "the square of its input". We could write this function as `y=x^2`. Then, when we give it an input like "3", the function takes that input, squares it, and returns "9". If we give it "3" again, it does the same thing, returning "9" again.

On the other hand, if we had a mapping like `y = +-sqrt x`, then `y` takes in any input (like "4"), finds the square root of this ("2"), then maps "4" to both `+2` and `-2`. This is not a function, because for every (positive) input, we get two outputs. A function requires "give one thing, get one thing".

Let's say we now have a mapping that takes whatever input we give it, and maps it to "6"—written as `y=6`. Is this still a function? Yes it is, because for each input (like "3"), the "computer program" takes the input, disregards it, and just returns "6". If we give it "3" again, we're still guaranteed to get "6" and only "6".

Bonus:

When you graph a mapping, the easiest way to tell if it's a function is to do the "vertical line test". If there's ever a vertical line that passes through the graph more than once, the mapping is not a function. Otherwise, it is. The graph of `y=6` passes this test:

graph{0x+6 [-11.25, 11.245, -1.99, 9.26]}

Whereas `y=+-sqrt x` does not:

graph{x=y^2 [-7.55, 14.945, -5.5, 5.75]}