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Mathematics

Functions

Easy

Question

The domain of the function f(x) = $fraction numerator 1 over denominator square root of vertical line x vertical line minus x end root end fraction text is : end text$

$(- \infty, 0)$
$(- \infty, \infty) - {0}$
$(- \infty, \infty)$
$(0, \infty)$

The correct answer is: $left parenthesis negative straight infinity comma space 0 right parenthesis$

$vertical line x vertical line minus x greater than 0 not stretchy rightwards double arrow x less than 0 not stretchy rightwards double arrow text domain end text equals left parenthesis negative straight infinity comma space 0 right parenthesis$

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$text If end text f colon R rightwards arrow R text and end text g colon R rightwards arrow R text are defined by end text f left parenthesis x right parenthesis equals x minus left square bracket x right square bracket text and end text g left parenthesis x right parenthesis equals left square bracket x right square bracket text for end text x element of R text , end text$ where [x] is the greatest integer not exceeding x, $text then for every end text x element of R comma f left parenthesis g left parenthesis x right parenthesis right parenthesis equals$

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$text If end text f left parenthesis x right parenthesis equals left square bracket x right square bracket comma space g left parenthesis x right parenthesis equals x minus left square bracket x right square bracket$ Then which of the following functions is the zero functions

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

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

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$text For end text x element of R minus left curly bracket 0 comma 1 right curly bracket comma less or equal than t f subscript 1 left parenthesis x right parenthesis equals 1 over x comma f subscript 2 left parenthesis x right parenthesis equals 1 minus x text and end text f subscript 3 left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 minus x end fraction$ be three given functions if a function,j(x) satisfies $open parentheses f subscript 2 ring operator J ring operator f subscript 1 close parentheses left parenthesis x right parenthesis equals f subscript 3 left parenthesis x right parenthesis comma then J left parenthesis x right parenthesis$ is equal to

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If [ ] denotes the greatest integer function and if $f colon left parenthesis 5 comma 10 right parenthesis rightwards arrow left parenthesis 7 comma 12 right parenthesis$ is a function defined by $f left parenthesis x right parenthesis equals x plus 2 open square brackets x over 5 close square brackets text , then end text$

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