Question

Show that the set

T ={(w,x,y,z)∈R4 such that y=w and x^2 =z^4}

is not the graph of any function of w and x.

Answer 1

well u have the correspondence (i don't know if this is the word in english)
\[ \large (w,x)\mapsto(y,z) \]
such that \(y=w\) and \(z^4=x^2\).

Answer 2

do u remember the definition of function?

Answer 3

A function is a rule that assigns a unique element in Rn to Rm. It is one to one.

Answer 4

the last part is something else.

Answer 5

a function assigns a UNIQUE element to EVERY element of its domain.

Answer 6

Ok I understand that.

Answer 7

So a certain w and x cannot have two outputs.

Answer 8

what does (w,x)=(1,-1) get assigned to?

Answer 9

yes. precisely.

Answer 10

(1,-1) is assigned to (1,1)?

Answer 11

just that?

Answer 12

let's see:
\[ \large y=w=1 \]
right?

Answer 13

right

Answer 14

BUT
\[ \large z^4=x^2=(-1)^2=1\Rightarrow z=\pm\sqrt[4]{1}=\pm1 \]

Answer 15

so
\[ \large (1,-1)\mapsto(1,1) \]
and
\[ \large (1,-1)\mapsto(1,-1) \]

Answer 16

So does that mean T is not a function?

Answer 17

yes. that's what u were asked to prove.

Answer 18

T is graph of something, but that something is not a function. right?

Answer 19

yes.

Answer 20

Thank you SO much!

Answer 21

u r welcome.