Is the Function f(x) = 2x/(x-1) Both Injective and Surjective?

[mtayab1994]

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?

 

[LCKurtz]

Sure. If y is in ]2,+∞[, calculate what x in ]1,+∞[ gives f(x) = y.

 

[Ray Vickson]

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

 

[mtayab1994]

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.

 

[Ray Vickson]

What does \Delta = 4 + 4y^2 have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV

 

[mtayab1994]

What does \Delta = 4 + 4y^2 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.

 

[Deveno]

What does \Delta = 4 + 4y^2 have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV

i believe he's taking the discriminant of a quadratic.

 

[Ray Vickson]

i believe he's taking the discriminant of a quadratic.

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

RGV