How to Prove a Function is Injective(one-to-one) Using the Definition

How to prove a function is injective. Injective functions are also called one-to-one functions. This is a short video focusing on the proof.

[mayankkhare]

In just 3 minutes, I gained more knowledge than I did in 4 hours of class. Thanks a lot

[iamplaceholder]

Could use more difficult examples. This example is the one literally everyone covers and doesn't help at all in figuring out difficult functions.

[daved]

These videos on Injunctive and surjunctive proofs were super helpful. Thanks.

[KingGisInDaHouse]

For those that want an approach that’s a little more obvious. f(x)=x^2. x>=0
in order for this to be on to one
f(x+h) must not equal f(x) in any other circumstance other than h=0 or h not being real.
f(x+h)= (x+h)^2=x^2+2xh+h^2
=f(x)+2xh+h^2 it’s not the same as f(x) unless 2xh+h^2 is zero
2xh+h^h=0
h(2x+h)=0
h=0 2x+h=0
h=-2x
There are points were f(x+h) = f(x).
Where h=-2x.
f(x-2x)=f(-x)
However since we said x>=0
X is positive -1x is just it’s opposite and it’s disqualified so we ignore this case it is one to one.

[jcasma]

Grear video man. Would have loved if you had included some numeric examples, but it's good nonetheless.

[jingyiwang]

Thank you for this amazing video! I was so confused with how to write a clear and precise proof for my homework. Now I am not confused anymore. Thanks!😀

[nuche]

Amazing video

How do you prove that f(a+b)=(a+b, a-b) is injective

[gautamganesh]

Thanks so much for this!! This cleared all my doubts!

[jonathoncliffbailey]

Hey man thanks a lot! A lot easier to follow than Khan Academy.

[ScottRachelson]

Can you do some proofs involving function spaces and not just individual functions?

[eriannewyvestarter]

But the graph is parabola and it's coming from two different x's. Injective is one-to-one, and the proof isn't satisfying the definition.

[emigames]

Problem: what about using contrapositives?

[theonewhoisfluff]

This ultimately proves that there are no two elements with same function and just 1 element with a unique image right?

[sky_island]

Thankyou! This explanation was excellent

[abhishekthakur]

IF R is a set of real numbers then show that the function f:R-> R defined by f(x) = -sin x, is neither one-one nor onto

[garbagecuber]

Literally so easy to understand! Great video!

[vicente]

how would you execute this proof for a piecewise function?

[lanl]

idk how my prof can make this shit a 4 hour presentation, thank you sir

[eriktruong]

a = b, if a=1 and b=2 would that mean 1=2?

[rick]

but if you graph y = x^2, there are more than one x values that correspond with the same y value????