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nicnicman
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Homework Statement
Is the function one-to-one, onto, both, or neither?
f: Z→Z has the rule of f(n) = 4n^3 - 1
Homework Equations
The Attempt at a Solution
My answer: one-to-one
Is this correct?
nicnicman said:One-to-one proof:
4u^3 - 1 = 4v^3 - 1
4u^3 = 4v^3
u^3 = v ^3
u = v
In mathematics, "1-1" is a term used to describe a function that maps each element of the domain to a distinct element in the range. This means that for every input, there is only one output. Another way to think of it is that no two elements in the domain have the same image in the range.
An "onto" function, also known as a surjective function, is a function that maps the entire domain to the entire range, meaning that every element in the range has at least one corresponding element in the domain. In other words, the range is equal to the co-domain of the function.
Yes, a function can be both "1-1" and "onto". This type of function is known as a bijection and it has a one-to-one correspondence between the domain and the range. This means that every element in the domain has a unique element in the range and every element in the range has a unique element in the domain.
The main difference between "1-1" and "onto" functions is that "1-1" functions have a one-to-one correspondence between the domain and the range, while "onto" functions have a one-to-one correspondence between the entire domain and the entire range. "1-1" functions may have elements in the range that are not mapped to, while "onto" functions must map all elements in the range.
To determine if a function is "1-1", you can use the horizontal line test, which states that if a horizontal line intersects the graph of the function at more than one point, then the function is not "1-1". To determine if a function is "onto", you can use the vertical line test, which states that if a vertical line intersects the graph of the function at more than one point, then the function is not "onto". If a function passes both the horizontal line test and the vertical line test, then it is both "1-1" and "onto". If it fails one or both tests, then it is neither "1-1" nor "onto".