a) $$ f(x)=\frac{|x|}{2 x^{10}} $$

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We begin with the function f(x) = |x|/(2·x¹⁰). Since the absolute value |x| behaves differently for positive and negative x, it is useful to rewrite the function in a piecewise form. Recall that |x| = x if x > 0, |x| = –x if x < 0. Thus, for x ≠ 0 we have: For x > 0: f(x) = x/(2·x¹⁰) = 1/(2·x⁹). For x < 0: f(x) = (–x)/(2·x¹⁰) = –1/(2·x⁹). So the piecewise definition is: f(x) = { 1/(2·x⁹) if x > 0, –1/(2·x⁹) if x < 0 }. Alternatively, this can be written in a single formula by using the signum function (sgn x), defined as sgn(x) = 1 if x > 0, sgn(x) = –1 if x < 0. Thus, for x ≠ 0 we have: f(x) = [sgn(x)]/(2·x⁹). Note that f(x) is undefined at x = 0. This completes the analysis of the given function. The function $$ f(x) = \frac{|x|}{2x^{10}} $$ can be written as: - For $$ x > 0 $$: $$ f(x) = \frac{1}{2x^9} $$ - For $$ x < 0 $$: $$ f(x) = -\frac{1}{2x^9} $$ The function is undefined at $$ x = 0 $$.

a) \( f(x)=\frac{|x|}{2 x^{10}} \)

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