Injective functions in mathematics are mappings where each input has a unique output, ensuring one-to-one correspondence between elements of the domain and range. This text explores how to identify injective functions using the horizontal line test, their behavior in compositions, and provides examples and counterexamples. Understanding the difference between injective, surjective, and bijective functions is also discussed, highlighting the importance of these concepts in mathematical studies.
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Injective functions are mappings where each input is paired with a unique output
The horizontal line test is a visual method to determine if a function is injective
The composition of two injective functions is also injective
Linear functions, such as f(x) = 2x + 1, are injective due to their unique output for each input
Logarithmic functions, such as f(x) = ln(x), are injective due to their unique output for each input
Certain polynomial functions, such as f(x) = x^3, are injective due to their unique output for each input
Quadratic functions, such as f(x) = x^2, are not injective because they can map different inputs to the same output
Trigonometric functions, such as f(x) = cos(x), are not injective over their entire domain because they can yield identical outputs for different inputs
Surjective functions cover every element in the range with at least one element from the domain
Bijective functions are both injective and surjective, establishing a perfect pairing between the domain and range elements
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Injective Function Example
Function assigning unique IDs to students; each student has a different number.
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Domain vs. Range in Injective Functions
Domain: set of all inputs. Range: set of all actual outputs. Injective maps one-to-one from domain to range.
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Injective Function Outcome
Every element of the range is paired with exactly one element of the domain.
