Question

Can someone give me one example about injective, surjective and biyective functions. I need to reinforce what I read.

Answer 1

First of all I will start by stating what the synonyms for the terms are:

Injective: one to one
Surjective: onto

Answer 2

A one to one function is such that
The image of every element in the domain yields a distinct point in the range.

Answer 3

Or formally:
If \(f(x)\) is one to one then
\[f(x_1) = f(x_2) \implies x_1 = x_2\]

Answer 4

An onto function is simply one where the image of the domain is the entire target space.

Answer 5

Finally bijective functions are functions that are both one to one and onto.

Answer 6

Formally a function \(f:X->Y\) is onto if:
For every \(y \in Y\) there is an \(x \in X\) such that \(f(x) = y\)

Answer 7

Any questions?

Answer 8

Yeah

Answer 9

Can you help me to determine if the following function is injective surjective, byyective or non of the above mencioned.

f(x)=1/(x2 +1)

Answer 10

no im helping my brother to get into the navy

Answer 11

that is one of my brother questions

Answer 12

Is it one to one?
No, it is an even function so
f(-x) = f(x)
Therefore it is not injective

Is it onto?
Is the image of the domain the set of all real numbers?
No it is not, so the function is not surjective.

Answer 13

Just to confirm, this is the function
\[f(x)=\frac{1}{x^2+1}\]correct?

Answer 14

Correct me if im wrong, but if I want to find out if a function is 1-2-1 I should find out if this function is even? right?

Answer 15

yeah, the function is correct

Answer 16

No, even functions are not one to one, but not all functions that are not one to one are even.

Answer 17

It is a sufficient condition, not a necessary one.

Answer 18

ok till now I understand. can you give me another example like the one we just worked on, please

Answer 19

Ok, for example, the function f(x)=\(x^3\) is both one to one and onto.

Answer 20

can you show the work to determine that :)

Answer 21

What is the image of the domain?

Answer 22

The function \(x^3\) diverges to negative infinity for large \(-x\) and positive infinity for large \(x\).

Answer 23

Then by the intermediate value theorem it will attain all the values between \(-\infty\) and \(\infty\). Therefore the image of the domain is \(\mathbb{R}\)

Answer 24

ok got it. Tanks alchemista

Answer 25

Also to prove its one to one, you need to show that if \(f(x_1)=f(x_2)\) then \(x_1 = x_2\)

Answer 26

Show that if \((x_1)^3 = (x_2)^3\) then \(x_1 = x_2\). That should be easy.

Answer 27

ok gonna try with other function later. Now I have to go to college. Thank you very much for you help. If I have any problem later Im gonna write on this threat . hope you dont mind?