Determining if a function is injective

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SUMMARY

The function f(x) = x³ - 5x² + 3x + 5 is not injective due to the presence of local extrema. Specifically, if a continuous function has a local maximum or minimum, it will take on the same value at different points, violating the definition of injectivity. This conclusion is based on the analysis of the function's critical points and their implications on the function's behavior.

PREREQUISITES
  • Understanding of calculus concepts, particularly local extrema
  • Familiarity with the definition of injective functions
  • Knowledge of polynomial functions and their properties
  • Ability to analyze continuity in functions
NEXT STEPS
  • Study the concept of local extrema in calculus
  • Learn how to determine injectivity of functions
  • Explore the properties of polynomial functions
  • Investigate the implications of continuity on function behavior
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Students of calculus, mathematicians analyzing function properties, and educators teaching function behavior and injectivity concepts.

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is the function x³-5x²+3x+5 injective. how can you tell
 
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Re: calculus

You can determine if the function has a local extremum. If a function is continuous and has, say, a maximum, then it is not injective since it assumes the same values on both sides of that point of maximum.
 

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