ANSWER: Domain of f is the set "(-\\infty,-1]\\bigcup [1,2)" ; range of f is the set "[0,+\\infty)"
EXPLANATION. If "x>2" , then the denominator of the fraction is positive and the numerator is negative. Therefore the fraction "\\frac{1-x^{2}}{x-2}<0" . If "x<2" , then the domain of the function f consists of x, for which "1-{ x }^{ 2 }\\le 0" , hence "x\\in\\left( - \\infty,-1 \\right ] \\bigcup\\left[1,+\\infty \\right )" . Thus, the domain of the function is the set "\\left \\{ (-\\infty,-1]\\bigcup [1,+\\infty) \\right \\} \\bigcap (-\\infty,2) =" "(-\\infty,-1]\\bigcup [1,2)"
The range of the function is the set "[0,+\\infty)" , since for any "a\\geqslant0" the equation f(x)=a has a solution ,that belongs to [1,2). This follows from equivalent equalities :"x\\in [1,2)\\quad ,\\sqrt { \\frac { 1-{ x }^{ 2 } }{ x-2 } } =a\\quad \\Leftrightarrow \\quad \\frac { 1-{ x }^{ 2 } }{ x-2 } ={ a }^{ 2 }\\quad \\Leftrightarrow { \\quad x }^{ 2 }+{ a }^{ 2 }x-2{ a }^{ 2 }-1=0\\" "\\Rightarrow x=\\frac { -{ a }^{ 2 } }{ 2 } +\\sqrt { \\frac { { a }^{ 4 } }{ 4 } +2{ a }^{ 2 }+1 } \\quad"
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