[Grade 12 Maths: Inverse] Relationship between inverse and original

[u/CaliPress123]

Is f(f⁻¹(x)) = x always true? I see things online saying that it must be true but for things like inverse trig e.g sin^-1(sin(4π/3))=–π/3??

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[u/Alkalannar]

You need f to be an injection for f^-1 to be well-defined.

Otherwise, there is a range restriction on f^-1.

This is equivalent to a domain restriction on f, and over the restricted domain, f is an injection, so f^-1 is well-defined, and that's how we define f^-1.

Example: For x², we restrict the domain to x >= 0, so we restrict the range of x^1/2 to y >= 0.

For sin(x), we want -pi/2 <= x <= pi/2, which means arcsin(x) is going to have a range of [-pi/2, pi/2], and so on.

[u/cheesecakegood]

math language sometimes obscures the issue, but we can think through it logically.

"function" means that any input has a very particular output (no probability stuff).

inverse function means you can do detective work to figure out what the input was from any output.

chaining them means, in theory, that if you plug in the input, find the output, then find the input you'll end up where you started... but wait!

a function is allowed to have identical outputs. the only thing a function says, is that the mapping of an input to an output has to be deterministic, so to speak. that is, the input must be transformed in a predictable way. who cares if the function is something like f(x) = 1, where it always returns 1, it's still a function. But obviously if I tell you "output was 1, what was the input?" you can't tell.

so you can see that not all functions can be inverted than re-verted, you need a nicer function.

simple example is just x^2. any input maps to a particular output. but inverted, you have to split it up into two functions, not just one, because together they aren't a function.

what, mathematically, do we call these "nicer functions" that can be inverted and re-inverted at will?

• "one-to-one" or "injective" where we don't fall into the simple trap above: inputs and outputs can freely swap without issue. there is some nice graphical stuff that can help you visualize this - but essentially, both f and f^-1 must pass vertical line tests

• the other issue that we sorta glossed over is that not all functions (to be a function) need to have infinite domains and ranges. otherwise, we get the opposite problem, where at some point in the chain we might get an input that just doesn't even exist. that's obviously a no-go. math speak calls this "surjective" or "onto". look up the fancy definition if you want more detail, but that's the concept.

these two things together is a "bijection"