How can I prove the monotonicity of a function?Given is the function f ( x ) = x 1 − x 2 .How can I prove that this function is monotonic and thus injective?

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How can I prove the monotonicity of a function?Given is the function   f  (  x  )  =      x          1      −              x        2            .How can I prove that this function is monotonic and thus injective?

Houston Ochoa · Accepted Answer

Step 1On the interval [0,1), the numerator x is increasing and the denominator       1    −          x      2       is decreasing. So f(x) is increasing on the interval.For   −  1  ≤  x  ≤  0, use the fact that f(x) is an odd function, so has symmetry across the origin. We conclude that f(x) is monotone on (-1, 1), the natural domain of definition.Step 2For other functions, a different approach, such as looking at the sign of the derivative, may be the appropriate one. The derivative should not be the automatic go-to technique, since the derivative may be complicated, and difficult to analyze.

kaltEvallwsr · Accepted Answer

Step 1Generally approach is to take the derivative of the function and check if the derivative is positive or negative. If positive, the function is increasing. If negative, the function is decreasing.Step 2For this function,       f    ′    (  x  )  =      1          (      1      −              x        2                    )                  3                      /                    2                     which is apparently positive on interval (-1, 1). So f(x) is an increasing function on (-1, 1).

How can I prove the monotonicity of a function? f(x)=x/(sqrt{1-x^2})

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