Question 939658
To have a {{{function}}}, the pairs, of the form (x,y),
must be such that each x is paired with only one y.
To have a {{{red(one-to-one)}}} {{{function}}} , you need pairs, of the form (x,y),
That form a function (see above),
and also such that each y is paired with only one x.
If you have a {{{one-to-one}}} {{{function}}} ,
reversing the pairs you get another one-to-one function.
Also, if reversing the pairs of a function you get another function,
both functions are one-to-one functions,
and we say that each of the two functions is the {{{inverse}}} of the other function.
That is a lot of {{{names}}} to remember, and I am sorry that you have to go through this. (I am also glad I don't have to remember those names any more).
 
Is the given set a function?
The first members of each pair form the set {1,2,3,4} .
Since the first numbers in the pairs are all different,
you do not have the same x shown twice, paired with different y's.
 
After you reverse all the ordered pairs, is the new set a function?
Reversing the pairs you end up with
{ (2,1) , (1,2) , (4,3) , (3,4) }.
Thee new first members of each pair form the set {2,1,4,3} .
Again, those x values are all different,
so there is no danger of having the same one paired with two different y values.
The new set is also a function.
 
The original set and the reversed set are both {{{one-to-one}}} {{{functions}}} .
 
MORE NAMES:
Maybe you were taught a {{{vertical-line-test}}} to find if a {{{relation}}} ,
meaning a set of {{{ordered-pairs(x,y)}}} , is a function,
and a {{{horizontal-line-test}}} to find if it is also a {{{one-to-one}}} {{{function}}} .
Those go like this:
1) You plot your ordered pairs as points on an x-y chart.
I plotted your points (circled in green) below.
{{{drawing(300,300,-0.5,4.5,-0.5,4.5,
grid(2),green(circle(1,2,0.1)),
green(circle(2,1,0.1)),green(circle(3,4,0.1)),
green(circle(4,3,0.1))
)}}}
2) If there is no way to "skewer" two or more of the points with a vertical line,
your relation is a {{{function}}} .
That is the {{{vertical-line-test}}} .
3) If there is also no way to "skewer" two or more of the points with a horizontal line,
your relation is a {{{one-to-one}}} {{{function}}} .
That is the {{{horizontal -line-test}}} .
 
NOTE: I do not believe they should make you memorize all those names in math class.
You have to memorize enough stuff in the other classes, and memorizing makes math boring.
I think math is a lot of fun, and I use all the time
it to solve real life problem,
and to satisfy my curiosity about puzzling stuff that is not a problem to me
(like the problems in this website).
Unfortunately, I am not a teacher,
and they would not let me be one,
because I am not good enough at boring students to tears.