Math Problem Statement
One-to-one function Definition Example/Symbol
Solution
Definition: One-to-One Function
A one-to-one function (or injective function) is a type of function where each element of the domain maps to a unique element of the codomain. In other words, no two different inputs in the domain produce the same output in the codomain. Mathematically, a function is one-to-one if:
This ensures that the function does not "overlap" values in its range.
Example and Symbol
Example:
The function is one-to-one because for any two distinct inputs and , .
Symbol:
One-to-one functions are often expressed symbolically using the implication: Alternatively, a function is not one-to-one if:
Visual Example:
- For (defined on ), the function is one-to-one.
- For (defined for all real ), it is not one-to-one because .
Would you like further clarification or a detailed proof of the example?
Relative Questions
- How do we determine if a function is one-to-one using derivatives?
- Can a one-to-one function also be onto? What is the relation?
- How can we test if a function is one-to-one using the horizontal line test?
- Are all linear functions one-to-one? Why or why not?
- How does restricting the domain of a function help make it one-to-one?
Tip: Use the horizontal line test: If a horizontal line intersects the graph of a function more than once, the function is not one-to-one.
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Math Problem Analysis
Mathematical Concepts
Functions
Injective Functions
One-to-One Functions
Formulas
f(x1) = f(x2) implies x1 = x2
Theorems
Horizontal Line Test
Suitable Grade Level
Grades 9-12
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