Injective Functions (and a Proof!) | Injections, One to One Functions, Injective Proofs

What is an injective function? How do we prove a function is injective? Sometimes called an injection, or a one-to-one function, we’ll be defining injective functions in today’s video math lesson! We’ll also go over some examples of injective functions, examples of functions that aren’t injective, and we’ll briefly go over two ways to prove that a function is injective!

Let f be a function with domain A. Then f is injective if and only if for any two elements x, y of A, f(x) = f(y) implies x = y. Or equivalently, x not equaling y implies f(x) does not equal f(y). In other words, injective functions preserve distinctness! If two elements of the domain are distinct, then their images in the codomain are also distinct! No element in the codomain of an injection gets mapped to more than once!

As a brief explanation of the calculus proof method mentioned in the video, if a differentiable function’s slope is always positive and we take two unequal domain values x and y, suppose without loss of generality that x is less than y. Then we know f(x) is less than f(y) because the slope of f is positive. This proves f is injective because it proves distinct domain elements have distinct images. Similar reasoning holds for a function whose slope is always negative.

SOLUTION TO PRACTICE PROBLEM:

We let f map the reals to the reals with f defined as f(x) = 3x^2. This function is not injective. Consider 2 and -2. We see f(2) = 3*2^2 = 12 = 3*(-2)^2 = f(-2). Thus, f is not injective.



I hope you find this video helpful, and be sure to ask any questions down in the comments!

********************************************************************
The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

Vallow Bandcamp: https://vallow.bandcamp.com/
Vallow Spotify: https://open.spotify.com/artist/0fRtu...
Vallow SoundCloud:   / benwatts-3  
********************************************************************

+WRATH OF MATH+

◆ Support Wrath of Math on Patreon:   / wrathofmathlessons  

Follow Wrath of Math on...
● Instagram:   / wrathofmathedu  
● Facebook:   / wrathofmath  
● Twitter:   / wrathofmathedu  

My Music Channel:    / seanemusic