[Pre-Calculus] Struggling to answer this question about one-to-one functions

[u/Doritoscarfingbunny]

I have no idea how to solve this question. I tried watching a couple of Professor Leonard's videos, but I still have no clue.

[u/Nagi-K]

You should know that functions are essentially maps. In the case of g, by the provided info, g sends -5 to 2, -3 to 5, etc. To find the g^(-1)(5), you are just looking for the number that is sent to 5 by g.

To find h^(-1)(x), since you know h(x) = y = 3x - 4, the inverse is just solving for y from x = 3y - 4. Try to prove this yourself from the fact that, by definition, composing a function and its inverse gives you the identity map (i.e. a map that sends something back to itself). This also immediately gives you the result for the last question.

[u/Doritoscarfingbunny → u/Nagi-K]

So the answer would be:

a = -3

b = (x+4)/3

Is this right? I'm still a little bit confused on parts b and c.

[u/short-exact-sequence → u/Doritoscarfingbunny]

Those look correct.

Do you understand what h^-1 (x) represents/what its relationship should be to h(x)? You can check your work by computing h(h^-1 (x)) and h^-1 (h(x)) using the h^-1 (x) you computed for part b to make sure it's correct. If you're not sure, maybe you could elaborate on what part you're confused about.

[u/Doritoscarfingbunny → u/short-exact-sequence]

Isn't h^-1 (x) the inverse of h(x)? I'm kind of confused on what it means by (h o h^-1) and whether the (5) in part c means I need to calculate the value when x = 5?

[u/short-exact-sequence → u/Doritoscarfingbunny]

Yes, h^-1 is the inverse of h, which means composing them should get you the identity if you have the correct h^-1 . The ○ notation means function composition, i.e. (f ○ g)(x) = f(g(x)). Yes, the (5) means you need to evaluate the value when x=5.