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Is the square of a function always positive

  1. Feb 9, 2012 #1
    I'm curious, is the square of any function always positive?

    It seems obvious that it's always positive because if you have a function (F), an input (x) and an output (y) then you have y = F(x)

    And if you square the function then (F(x))2 = y2 which means that every value in the range is now positive.

    Is this always true?

    Also, if you can also answer this that would be great, if I have two positive functions and I am taking the definite integral of the product of those two functions, will I always get a number greater than or equal to 0?
     
  2. jcsd
  3. Feb 9, 2012 #2

    HallsofIvy

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    Yes, what you said is correct if you are careful with your qualifiers. If f is a real valued function, then [itex]f(x)^2[/itex] is non-negative (not, strictly speaking, "always positive"). That is because, for any x, f(x) is just a real number and the square of any real number is non-negative. (Note the difference between the function "f", and the real number, "f(x)", which is a number.)

    If, for all x in [a, b], both f(x) and g(x) are greater than or equal to 0, then [itex]\int_a^b f(x)g(x)dx\ge 0[/itex].
     
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