- #1
Cha0t1c
- 15
- 5
- Homework Statement
- Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1. Prove f is injective.
- Relevant Equations
- f(a) = f(b) ==> a=b
Hello,
Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1.
I am trying to prove f is injective.
Let a,b be in ]1, +inf[ and suppose f(a) = f(b).
Then, a^2 + 2a + 1 = b^2 + 2b + 1.
How do I solve this equation such that I end up with a = b?
Solution:
(a + 1) ^2 = (b + 1)^2
sqrt[(a+1)^2] = sqrt[(b+1)^2]
abs(a + 1) = abs(b + 1)
since a>1 and b>1
a + 1 = b +1
thus a = b
hence f is injective.
[Moderator's note: Moved from a technical forum and thus no template.]
Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1.
I am trying to prove f is injective.
Let a,b be in ]1, +inf[ and suppose f(a) = f(b).
Then, a^2 + 2a + 1 = b^2 + 2b + 1.
How do I solve this equation such that I end up with a = b?
Solution:
(a + 1) ^2 = (b + 1)^2
sqrt[(a+1)^2] = sqrt[(b+1)^2]
abs(a + 1) = abs(b + 1)
since a>1 and b>1
a + 1 = b +1
thus a = b
hence f is injective.
[Moderator's note: Moved from a technical forum and thus no template.]
Last edited: