Solve Continuous Functions Equation: (f(x)^2)= x^2

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SUMMARY

The discussion centers on identifying continuous functions f that satisfy the equation (f(x)^2) = x^2 for all x. The key insight is derived from taking the square root, leading to the conclusion that |f(x)| = |x|. This results in two possible continuous functions: f(x) = x and f(x) = -x. Thus, there are exactly two continuous functions that meet the criteria of the equation.

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  • Understanding of continuous functions
  • Familiarity with absolute value properties
  • Basic knowledge of algebraic equations
  • Experience with function analysis
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  • Study the implications of absolute value in function equations
  • Learn about piecewise functions and their applications
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Students studying calculus, mathematicians interested in function properties, and educators teaching continuous functions and their characteristics.

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Homework Statement


How many continuous functions f are there which satisfy the equation (f(x)^2) = x^2 for all x?

Homework Equations


The Attempt at a Solution


What method should I use to solve this? Is there a specific strategy involved besides plug and chug? Off the top of my head, I can only think of f(x)= x, but that's just one. Thank you.
 
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If you take square roots:
[tex]\sqrt{ a^2} = |a|[/tex]
Is always a true statement regardless of whether a is positive or negative. So for your equation you get |f(x)| = |x|. Try working from there
 
Thank you so much for that derivation!
For this absolute function, there would be two possible values: -x and x.
There would only be two functions possible then I think?
I thought it would be more complicated, and involve log functions, etc.
 

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