1,000,000 Question Thread :D

[kenhung123]

Can someone explain how do you know whether the composition of functions are defined or not?
E.g. and

[TrueTears]

Can someone explain how do you know whether the composition of functions are defined or not?
E.g. and

If you want then

If you want then

If your composite function does not satisfy the above, then it is not defined.

[m@tty]

Toothpaste made a very good post about composite functions here. 

Follow the link in the quote 

[Souljette_93]

Oh, Thank God for this thread. i had problems with compositions this morning.


TC,
Souljette<3

[QuantumJG]

Can someone explain how do you know whether the composition of functions are defined or not?
E.g. and

I haven't done these in ages.

Ran(f) = [-0.25,infinity)
Ran(g) = [0,infinity)

so for f o g to exist, the ran(g) must be a subset of dom(f), which is true.

    for g o f to exist, the ran(f) must be a subset of dom(g), which is is false, since the ran(f)         is NOT a subset of the dom(g).

If you think carefully about sets it makese sense.

f states that it can map x onto the real numbers as long as it's a real number
(basically why you may see R -> R, that is what the function is mapping the set of x-values to)

g states that it can map x onto the real numbers as long as it's a positive real number or 0.

g can't map f onto the real numbers, since not all values of f are positive. 

 

[Hielly]

hey i have a composite question
f : {x: x ≤ 3} → R, f (x) = 3 − x and g: R → R, g(x) = x2 − 1
a) Show that f ◦ g is not defined.
b) Define a restriction g* of g such that f ◦ g* is defined and find f ◦ g*.

I've done a), im stuck on b) im not sure how the restriction works.

Thanks

[TrueTears]

Now we require .

We need to be defined.





So is not defined at the moment.

Thus we need to restrict to

So if then or

[kenhung123]

What does find f 0 h(x) mean?

[TrueTears]

[kenhung123]

TT can you explain how you derived that?

[/0]

lol you really love  

[TrueTears]

TT can you explain how you derived that?

It is not a deviation, it is just a matter of notation.

[QuantumJG]

hey i have a composite question
f : {x: x ≤ 3} → R, f (x) = 3 − x and g: R → R, g(x) = x2 − 1
a) Show that f ◦ g is not defined.
b) Define a restriction g* of g such that f ◦ g* is defined and find f ◦ g*.

I've done a), im stuck on b) im not sure how the restriction works.

Thanks

Simple!

Use your knowledge that for f o g to exist, ran(g) must be a subset of Dom(f).

Dom(f) is x <= 3, thus, find the values of x that makes g <= 3 ! Which occurs when -2 <= x <= 2, thus, you need to restrict Dom(g) to [-2,2].

 

[NE2000]

lol you really love  

yeah lol notice that too, did you use it on your exams?

[TrueTears]

lol you really love  

yeah lol notice that too, did you use it on your exams?

lol nah I only started using it heaps when I saw kamil using it in his explanations