Polynomial Functions.

  • Thread starter mtayab1994
  • Start date
  • #1
584
0

Homework Statement



f:]1,+∞[→]2,+∞[
x→ 2x/(x-1)


Homework Equations


Prove that f is injective and serjective.


The Attempt at a Solution



I already proved that it's injective by stating the injectivity law:
for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b

so: 2a/(a-1)=2b/(b-1) entails: 2ab-2b=2ab-2a entails -2b=-2a entails a=b

Can anyone please tell me how to prove that its serjective?
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,557
767
Sure. If y is in ]2,+∞[, calculate what x in ]1,+∞[ gives f(x) = y.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement



f:]1,+∞[→]2,+∞[
x→ 2x/(x-1)


Homework Equations


Prove that f is injective and serjective.


The Attempt at a Solution



I already proved that it's injective by stating the injectivity law:
for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b

so: 2a/(a-1)=2b/(b-1) entails: 2ab-2b=2ab-2a entails -2b=-2a entails a=b

Can anyone please tell me how to prove that its serjective?

In your own words, what does it mean to say that f is surjective? (That is sUrjective, not sErjective!) Turn that verbal statement into an equation and then work on the equation, to see what conclusions you can make, or else use some known, general properties to get a conclusion.

RGV
 
  • #4
584
0
i got it
f(x)=y
y=2x/x-1 equivalence y(x-1)=2x equivalence yx-2x-y=0
now we find Δ
Δ=4+4y^2
since Δ≥0 therefore there is some solution to this equation and therefore f is serjective.
 
  • #5
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
What does [itex] \Delta = 4 + 4y^2 [/itex] have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV
 
  • #6
584
0
What does [itex] \Delta = 4 + 4y^2 [/itex] have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV

Well my teacher stated that if we find that Δ≥0 then therefore f is surjective and btw my first language is english, but i'm learning overseas in Morocco and all the lessons here are in Arabic, so that's probably the reason why i spelled it wrong.
 
  • #7
Deveno
Science Advisor
906
6
What does [itex] \Delta = 4 + 4y^2 [/itex] have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV

i believe he's taking the discriminant of a quadratic.
 
  • #8
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
i believe he's taking the discriminant of a quadratic.

Of course he is; but where is the quadratic equation in this question?

RGV
 

Related Threads on Polynomial Functions.

  • Last Post
Replies
1
Views
831
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
837
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
17K
Replies
2
Views
1K
Top