Proof including a one to one function

In summary, a one to one function is a type of function where each input has a unique output, also known as an injective function. To prove that a function is one to one, you can use the horizontal line test, where if a horizontal line intersects the graph at more than one point, the function is not one to one. One to one functions are important in mathematics for their useful properties and real-world applications. A function can be both one to one and onto, known as a bijection. To prove that a function is one to one using algebraic methods, one can use the definition of one to one functions or algebraic manipulation to show that the inputs and outputs are unique and cannot be equal to each other.
  • #1
aesailor
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Homework Statement



if f(x) is a one-to-one function, then g(x) = f(x^3) is also a one-to-one function

Homework Equations





The Attempt at a Solution



Assume f(x) is a one to one function. For a particular element of B, there is at most one element of A which is mapped to it. Therefore, by the definition of injective x^3 must also be one to one.

This is what I have, but I don't think that this is the way to go about it.
 
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  • #2
If g(a)=g(b), then?
 

1. What is a one to one function?

A one to one function is a type of function where each input has a unique output. In other words, no two inputs can have the same output. This is also known as an injective function.

2. How can you prove that a function is one to one?

To prove that a function is one to one, you can use the horizontal line test. This test involves drawing a horizontal line across the graph of the function. If the line intersects the graph at more than one point, then the function is not one to one. If the line only intersects the graph at one point, then the function is one to one.

3. What is the importance of one to one functions in mathematics?

One to one functions are important in mathematics because they have many useful properties, such as being able to find an inverse function, which is useful for solving equations. They are also used in many real-world applications, such as in computer science and economics.

4. Can a function be both one to one and onto?

Yes, a function can be both one to one and onto. A function that is both one to one and onto is called a bijection. This means that each input has a unique output and every element in the output has a corresponding input.

5. How do you prove that a function is one to one using algebraic methods?

To prove that a function is one to one using algebraic methods, you can use the definition of one to one functions. This means showing that if f(a) = f(b), then a = b. You can also use algebraic manipulation to show that the inputs and outputs are unique and cannot be equal to each other.

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