How is f(x)=sqrt(x) a valid function?

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The square root function f(x) = sqrt(x) is considered valid because it adheres to the definition of a function, which states that each input must have a single output. While the equation a^2 = x yields both positive and negative solutions, the square root function specifically refers to the positive solution, denoted as sqrt(x). This convention is not an oversight but a standard practice to maintain the function's validity. Thus, sqrt(x) is defined as the positive number a such that a^2 = x, effectively providing a unique output for each input. The discussion clarifies that this approach does not ignore solutions but rather specifies the principal square root.
CuriousBanker
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My teacher has told me the square root function is a valid function. He has also told me that a function cannot possibly have two different output for one given input. 36^1/2 for instance has both -6 and +6 as answers. He told me to just refer to the positive square root...eh, that seems kind of sloppy to just ignore half of the answers out of convenience, no?
 
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is defined to be the positive number such that . It isn't ignoring half the answers unless you are asked for the numbers that solve the equation and give the answer instead of .
 
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