Algebra.Com's Answer #838860 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Find the range of f(x)= sqrt(x^2-2) + sqrt((1+x^2)/(2+x^2)) - (8-x^2)/(x^2-4x-1)
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document.write( "Our function f(x) is the sum of three addends p(x) =  , q(x) =  and r(x) =  . \r\n" );
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document.write( "Their plots are shown in figures 1, 2 and 3.\r\n" );
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document.write( "  \r\n" );
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document.write( " Figure 1. Plot p(x) = \r\n" );
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document.write( " Figure 2. Plot q(x) = \r\n" );
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document.write( "  \r\n" );
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document.write( " Figure 3. Plot r(x) = \r\n" );
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document.write( "  \r\n" );
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document.write( " Figure 4. Plot f(x) = + -  \r\n" );
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document.write( "From Figure 1 and from the formula for p(x), is is easily seen that p(x) is defined outside of {  < x <  },\r\n" );
document.write( "and so (hence) f(x) is also defined outside of this interval.\r\n" );
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document.write( "On the left of this interval, the range of p(x) is unbounded from 0 to infinity in positive direction.\r\n" );
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document.write( "From Figure 2 and from the formula for q(x), it is clear that function q(x) is defined on the whole number line \r\n" );
document.write( "and is continuous function bounded in infinity (both in positive and negative infinity), so this function \r\n" );
document.write( "is bounded continuous function.\r\n" );
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document.write( "From Figure 3 and from the formula for r(x), it is clear that the range of function r(x) is unbounded in negative \r\n" );
document.write( "direction in vicinity of its pole, which is located at the zero of its denominator x =  =  = 4.24.... *)\r\n" );
document.write( "At the same time, as x goes to positive or negative infinity, function r(x) remains bounded.\r\n" );
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document.write( "Collecting all these properties and making a plot of the function f(x) in Figure 4, we see that\r\n" );
document.write( "all these listed properties make the range of function f(x) as whole number line without any holes.\r\n" );
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document.write( "Solved.\r
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document.write( "*) This pole x = = 4.24.... does make a hole in the domain
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document.write( " of the function f(x), but does not make a hole in its range.\r
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