Functions’ Inverses

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Misc

End-of-Semester

The final day to grade anything for this class is Thursday May 20 (the last day of finals). Office hours end at 5:00 that day, like any other day.

I expect the week or so leading up to this deadline to be very busy. Work that’s ungraded at the end of it will get grades of 0. So if you have grading to catch up on, or “redos” that you want graded, start acting now to get it done, and do not count on being able to do it during finals week or study day.

Math Alumni Talk

Our AWM (Association for Women in Math) chapter is holding a panel with some Geneseo math alumni talking about their educations and careers as women mathematicians.

Next Wednesday (April 28), 5:00 - 6:00PM

Via Zoom: Meeting ID 931 4225 7839, passcode 237493

Functions’ Inverses

Based on section 6.5 and this discussion of inverses.

Functions and Ordered Pairs

Suppose f : ℤ → ℤ by f(n) = n². What are some ordered pairs in f?

All the pairs consisting of an integer n and the corresponding f(n).

So f = { ..., (1,1), (2,4), (-2,4), ( n, n² ), .... }

Suppose g : ℤ → ℤ = { ..., (-1,0), (0,1), (1,2), (2,3), ... }. What is another plausible definition of g?

Reading the pairs as examples, it looks like g(n) = n + 1.

What are some ordered pairs in g^-1? In f^-1?

f^-1 = { ..., (1,1), (4,2), (4,-2), ( n², n ), ... }. In other words, pairs of the form ( f(n), n ). In this case, this set is the inverse of f, even though it’s not a function (because it associates two different values, 2 and -2, with 4), i.e., inverses of functions always exist, but they aren’t always functions.

g^-1 = { ..., (0,-1), (1,0), ... }. This is a function (though the 2 example pairs shown don’t prove that, of course).

Theorems about Inverses

Which of the following functions from ℤ to ℤ have inverses that are also functions from ℤ to ℤ?

• f(n) = ⌊ n/2 ⌋. If m = ⌊ n/2 ⌋, then n = 2m or n = 2m+1. So f^-1 has to associate every integer with 2 others; not a function.
• h(n) = n + 7, if n + 7 = m, then n = m - 7, so h^-1(m) = m - 7, which is a function.
• g(n) = 2n + 1, if 2n + 1 = m, then n = (m-1)/2, so g^-1(m) = (m-1)/2, but this is not a function from all integers to integers.
• k(n) = n³ - 1, k^-1(m) = (m+1)^1/3. Not a function from ℤ to ℤ, like above.

A useful and possibly new notation: ⌊ x ⌋, the “floor” of x, is the largest integer less than or equal to x. For example ⌊ 1.3 ⌋ = 1, ⌊ 1.999 ⌋ = 1, ⌊ 2 ⌋ = 2, etc.

A random thought that occurred to me as I was trying to think of examples of invertible and not invertible functions for above: If f is a differentiable function from ℝ to ℝ, and f^-1 is also a function from ℝ to ℝ, what, if anything, do you know about (f’)^-1?

A good way to start working with questions such as this is to look at some concrete examples of what they’re talking about. So take a simple example, f(x) = x - 2. Then f^-1(y) = y + 2, which is a function, and f’(x) = 1. But (f’)^-1 is definitely not a function, since it has to associate 1 with every real number!

So maybe the rule is that differentiable, invertible, functions don’t have invertible derivatives?

It seems a little hasty to conclude that based on just one example. Maybe we should instead try to find an example that goes the other way, i.e., a differentiable and invertible function whose derivative is also invertible.

We left this as an exercise for the listener, so that we could consider one last point about invertible functions…

What did all the invertible functions above have in common? They were all bijections.

Key Points

Functions are sets of ordered pairs from dom(f) × codom(f), subject to the following rules:

1. If (a,b) and (a,c) are in f, then b = c (i.e., there can never be 2 pairs with the same first component but different second components)
2. For all a in dom(f), (a,x) is in f for some x in codom(f) (i.e., the function has to map every element of its domain)

It’s always the case that dom( f^-1 ) = codom( f ), i.e., a function’s inverse has to be able to map everything in that function’s codomain.

Function f is invertible (i.e., has a function as its inverse) if and only if f is a bijection.

Next

Relations, a generalization of the ordered pair view of functions.

Please read section 7.1 in the textbook.

Please also contribute to this discussion of relations.

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