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Prove this equation is one to one

  • Thread starter Painguy
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  • #1
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Homework Statement


[tex]y=(x+1)^3[/tex]


Homework Equations


f(a)=f(b)
a=b


The Attempt at a Solution


[tex](a+1)^3 =(b+1)^3[/tex]
[tex]1+3 a+3a^2+a^3=1+3b+3b^2+b^3[/tex]

This is where I get stuck.
 

Answers and Replies

  • #2
SammyS
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How about: if [itex](a+1)^3 =(b+1)^3\,,[/itex] then [itex](a+1)^3-(b+1)^3=0\,.[/itex]

Then factor into difference of cubes.
 
  • #3
Dick
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Do you know any calculus? That's the easy way to prove this. Show y is an always increasing function of x.
 
  • #4
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How about: if [itex](a+1)^3 =(b+1)^3\,,[/itex] then [itex](a+1)^3-(b+1)^3=0\,.[/itex]

Then factor into difference of cubes.
I know what the difference of cubes is but I can't seem to figure out how to apply that to this case. I was under the impression that if I was to apply the difference of cubes that the equation would gabber to be in, the form [tex](x^3)+1[/tex] rather than [tex](x+1)^3[/tex] I'm probably forgetting something here :P

Do you know any calculus? That's the easy way to prove this. Show y is an always increasing function of x.
I haven't taken calculus yet but would it suggestion look like this? f(x)=(x+1)^3 I've already graphed it & seen that it's 1:1 but I'm looking to do this algebraically. Excuse me if I misinterpreted your post.
 
  • #5
SammyS
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The difference of cubes is:

p3 - r3 = (p-r)(p2+pr+r2)

If (p-r)(p2+pr+r2) = 0,

Then either p=r,

or [itex]\displaystyle p=\frac{-r\pm\sqrt{r^2-4r^2}}{2}[/itex], which is only a real number if r = 0.
 
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  • #6
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m3 - n3 = (m - n)(m2 + mn + n2)

m3 + n3 = (m + n)(m2 - mn + n2)
 
  • #7
Deveno
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one could approach this another way:

if f:A→A is injective, and g:A→A is injective, then fog:A→A is injective.

proof:

suppose fog(x) = fog(y), that is:

f(g(x)) = f(g(y)).

since f is injective, g(x) = g(y).

since g is injective, x = y, QED.

now prove f(x) = x3 is injective, and

g(x) = x+1 is injective.
 
  • #8
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another path, i think the easiest mechanically but the least intuitive if you're given another definition, is to use the logical contrapositive equivalent of 1-to-1 (perhaps better known as proof by contradiction):

"x != y implies g o h(x) != g o h(y)".

here you don't need to show 1-to-1 of anything, only that the functions g and h are defined at x and y as you construct the conclusion f= g o h.

[edit] of course that ends up needing to show that g and h are indeed functions...
 
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