Proving 1-1 Functions with f(a) = f(b) and a = b: x^3-2 and x^4+2

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In summary, for [1] and [2], the equations have one and two solutions, respectively, but for [3], there is no solution.
  • #1
jwxie
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Homework Statement



[1] f(x) = x^3-2
[2] f(x) = x^4+2

Homework Equations



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

The Attempt at a Solution


Assuming that f(a) = f(b), for which some a =/ b

So for [1] f(x) = x^3-2, let assume a=/b, that f(a) = f(b)
in the end I got, a^3 = b^3

for [2], I did the same thing, and for f(x) = x^4+2, in the end I got a^4 = b^4

Now I need to simply them down to a = b, so I think I need to take cubic root on both side, and square root on both side, respectively.

Now why is [2] not a 1-1 function, while [1] is a 1-1 function?
When I solve for x, for example, in the same of y = x^2, i get sqrt of y = x, so it is not a 1-1 function.

What about x^3-2? I got cub root of y + 2 = x
Thanks
 
Last edited:
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  • #2
jwxie said:

Homework Statement



[1] f(x) = x^3-2
[2] f(x) = x^4+2


Homework Equations



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


The Attempt at a Solution


Assuming that f(a) = f(b), for which some a =/ b

So for [1] f(x) = x^3-2, let assume a=/b, that f(a) = f(b)
in the end I got, a^3 = b^3

for [2], I did the same thing, and for f(x) = x^4+2, in the end I got a^4 = b^4

Now I need to simply them down to a = b, so I think I need to take cubic root on both side, and square root on both side, respectively.
The first one is usually called the cube root. For the other you need a fourth root, not a square root.

You don't need the cube root and fourth root, though. In fact, it's best to not use them. For the first problem you have
a3 = b3 <==> a3 - b3 = 0.
The last equation can be factored to give three solutions, two of which are complex.

For the other problem, you have
a4 = b4 <==> a4 - b4 = 0.
This equation can be factored to give four solutions, two of which are complex.

You can ignore the complex solutions. Can you find the real solutions to these equations?

jwxie said:
Now why is [2] not a 1-1 function, while [1] is a 1-1 function?
I mean sometime cubic root can have -1, -1, 1..

Thanks
 
  • #3
Hi, thanks.
a3 = b3 <==> a3 - b3 = 0.
The last equation can be factored to give three solutions, two of which are complex.

For the other problem, you have
a4 = b4 <==> a4 - b4 = 0.
This equation can be factored to give four solutions, two of which are complex.

What I do not understand is how can I tell
When I solve for x, for example, in the same of y = x^2, i get sqrt of y = x, so it is not a 1-1 function.

What about x^3-2? I got cub root of y + 2 = x

For your question, I didn't really think about those complex number (since I didn't even know there are complex numbers in those roots).
I guess, from your information, you just literally pointed out that [1] there is only one a (b), and [2] there is two a, b that can still produce 0 (the value of y).
 
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  • #4
Can you solve the equation a4 - b4 = 0 by factoring? This is the same as ((a2)2 - (b2)2 = 0.

Can you solve the equation a3 - b3 = 0 by factoring?
 
  • #5
[itex](-1)^3\no 1^3[/itex]

[itex](-1)^4 = 1^5[/itex]
 

What is a 1-1 function?

A 1-1 function, also known as an injective function, is a type of mathematical function where each element in the domain is paired with a unique element in the range. This means that for every input, there is only one corresponding output. In other words, no two elements in the domain can have the same output.

How do you determine if a function is 1-1?

To determine if a function is 1-1, you can use the horizontal line test. If a horizontal line can intersect the graph of the function at more than one point, then the function is not 1-1. Additionally, you can also check if each element in the domain has a unique output in the range by using the vertical line test. If a vertical line intersects the graph at more than one point, the function is not 1-1.

What is the difference between a 1-1 function and a many-to-one function?

The main difference between a 1-1 function and a many-to-one function is that a 1-1 function has a unique output for every input, while a many-to-one function may have multiple inputs with the same output. In other words, a 1-1 function is a special case of a many-to-one function where no two inputs have the same output.

Can a function be both 1-1 and onto?

Yes, a function can be both 1-1 and onto. A function that is both 1-1 and onto is called a bijection. This means that every element in the domain is paired with one and only one element in the range, and every element in the range has at least one element in the domain that maps to it.

What is the significance of a 1-1 function?

1-1 functions have many applications in mathematics, engineering, and science. They are important in creating mathematical models, solving equations, and understanding relationships between different variables. In computer science, 1-1 functions are used in encryption and data compression algorithms. Additionally, in statistics, 1-1 functions are used to transform data to make it easier to analyze and interpret.

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