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Mathematics

Functions

Easy

Question

If $f left parenthesis x right parenthesis equals fraction numerator sin 3 x over denominator sin x end fraction comma x not equal to n pi$ then the range of f(x) is

$[- 1, 3]$
$(- \infty, - 1)$
$(3, \infty)$
$[- 1, 3)$

The correct answer is: $left square bracket negative 1 comma 3 right parenthesis$

$text If end text f left parenthesis x right parenthesis equals y text then end text y equals 3 minus 4 sin squared x not stretchy rightwards double arrow sin squared x equals fraction numerator 3 minus y over denominator 4 end fraction x not equal to n pi not stretchy rightwards double arrow 0 less than sin squared x less or equal than 1 not stretchy rightwards double arrow 0 less than fraction numerator 3 minus y over denominator 4 end fraction less or equal than 1 not stretchy rightwards double arrow 0 less than 3 minus y less or equal than 4 not stretchy rightwards double arrow negative 3 less than negative y less or equal than 1 not stretchy rightwards double arrow negative 1 less or equal than y less than 3$

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$text If end text f left parenthesis x right parenthesis equals fraction numerator x over denominator square root of 1 minus x squared end root end fraction comma space g left parenthesis x right parenthesis equals fraction numerator x over denominator square root of 1 plus x squared end root end fraction text then end text left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals$

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$text If end text f left parenthesis x right parenthesis equals sin squared x plus sin squared left parenthesis x plus pi divided by 3 right parenthesis plus cos x cos left parenthesis x plus pi divided by 3 right parenthesis space a n d space g left parenthesis 5 divided by 4 right parenthesis equals 1 text then end text open parentheses g o f close parentheses left parenthesis x right parenthesis equals$

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