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Mathematics

Functions

Easy

Question

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

$(f + g) (x)$
$(f g) (x)$
$(f - g) (x)$
$(f o g) (x)$

The correct answer is: $open parentheses f o g close parentheses left parenthesis x right parenthesis$

$left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals f left square bracket g left parenthesis x right parenthesis right square bracket equals f left parenthesis x minus left square bracket x right square bracket right parenthesis equals f left parenthesis left curly bracket x right curly bracket right parenthesis equals left square bracket left curly bracket x right curly bracket right square bracket equals 0 comma text since end text 0 less or equal than left curly bracket x right curly bracket less than 1$

Related Questions to study

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

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

MathematicsFunctions

$text For end text x element of R minus left curly bracket 0 comma 1 right curly bracket comma less or equal than t f subscript 1 left parenthesis x right parenthesis equals 1 over x comma f subscript 2 left parenthesis x right parenthesis equals 1 minus x text and end text f subscript 3 left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 minus x end fraction$ 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

$text For end text x element of R minus left curly bracket 0 comma 1 right curly bracket comma less or equal than t f subscript 1 left parenthesis x right parenthesis equals 1 over x comma f subscript 2 left parenthesis x right parenthesis equals 1 minus x text and end text f subscript 3 left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 minus x end fraction$ 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

MathematicsFunctions

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

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

MathematicsFunctions

parallel

$text If end text f colon left square bracket negative 6 comma 6 right square bracket not stretchy rightwards arrow R text is defined by end text f left parenthesis x right parenthesis equals x squared minus 3 text for end text x element of R text then end text open parentheses f space o space f space o space f close parentheses left parenthesis negative 1 right parenthesis plus open parentheses f space o space f space o space f close parentheses left parenthesis 0 right parenthesis plus open parentheses f space o space f space o space f close parentheses left parenthesis 1 right parenthesis equals$

$text If end text f colon left square bracket negative 6 comma 6 right square bracket not stretchy rightwards arrow R text is defined by end text f left parenthesis x right parenthesis equals x squared minus 3 text for end text x element of R text then end text open parentheses f space o space f space o space f close parentheses left parenthesis negative 1 right parenthesis plus open parentheses f space o space f space o space f close parentheses left parenthesis 0 right parenthesis plus open parentheses f space o space f space o space f close parentheses left parenthesis 1 right parenthesis equals$

MathematicsFunctions

$text If end text f left parenthesis x right parenthesis equals open parentheses a minus x to the power of n close parentheses to the power of 1 divided by n end exponent comma text where end text a greater than 0 text and end text n element of N comma text then end text open parentheses f subscript 0 f close parentheses left parenthesis x right parenthesis equals$

$text If end text f left parenthesis x right parenthesis equals open parentheses a minus x to the power of n close parentheses to the power of 1 divided by n end exponent comma text where end text a greater than 0 text and end text n element of N comma text then end text open parentheses f subscript 0 f close parentheses left parenthesis x right parenthesis equals$

MathematicsFunctions

$text If end text F left parenthesis n right parenthesis equals left parenthesis negative 1 right parenthesis to the power of n minus 1 end exponent left parenthesis n minus 1 right parenthesis text and G end text left parenthesis n right parenthesis equals n minus F left parenthesis n right parenthesis space straight for all n element of N text then end text open parentheses G subscript 0 G close parentheses left parenthesis n right parenthesis equals$

$text If end text F left parenthesis n right parenthesis equals left parenthesis negative 1 right parenthesis to the power of n minus 1 end exponent left parenthesis n minus 1 right parenthesis text and G end text left parenthesis n right parenthesis equals n minus F left parenthesis n right parenthesis space straight for all n element of N text then end text open parentheses G subscript 0 G close parentheses left parenthesis n right parenthesis equals$

MathematicsFunctions

parallel

$text If end text g left square bracket f left parenthesis x right parenthesis right square bracket equals vertical line sin x vertical line comma space f left square bracket g left parenthesis x right parenthesis right square bracket equals left parenthesis sin square root of x right parenthesis squared text then end text$

$text If end text g left square bracket f left parenthesis x right parenthesis right square bracket equals vertical line sin x vertical line comma space f left square bracket g left parenthesis x right parenthesis right square bracket equals left parenthesis sin square root of x right parenthesis squared text then end text$

MathematicsFunctions

$text If end text f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis squared minus 1 comma x greater or equal than negative 1 text then the set end text S equals open curly brackets x element of R colon f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent x close curly brackets text is end text$

$text If end text f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis squared minus 1 comma x greater or equal than negative 1 text then the set end text S equals open curly brackets x element of R colon f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent x close curly brackets text is end text$

MathematicsFunctions

$text Let end text x not equal to 0 comma vertical line x vertical line less than 1 half text and end text f left parenthesis x right parenthesis equals 1 plus 2 x plus 4 x squared plus 8 x cubed plus horizontal ellipsis. text Then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

$text Let end text x not equal to 0 comma vertical line x vertical line less than 1 half text and end text f left parenthesis x right parenthesis equals 1 plus 2 x plus 4 x squared plus 8 x cubed plus horizontal ellipsis. text Then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

MathematicsFunctions

parallel

$text If end text f colon left square bracket 1 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 1 comma straight infinity right square bracket text is defined by end text f left parenthesis x right parenthesis equals fraction numerator 1 plus square root of 1 plus 4 log subscript 2 x end root over denominator 2 end fraction text then end text f to the power of negative 1 end exponent left parenthesis 3 right parenthesis equals$

$text If end text f colon left square bracket 1 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 1 comma straight infinity right square bracket text is defined by end text f left parenthesis x right parenthesis equals fraction numerator 1 plus square root of 1 plus 4 log subscript 2 x end root over denominator 2 end fraction text then end text f to the power of negative 1 end exponent left parenthesis 3 right parenthesis equals$

MathematicsFunctions

$text If end text f left square bracket 1 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 1 comma straight infinity right parenthesis text is defined by end text f left parenthesis x right parenthesis equals 2 to the power of x left parenthesis x minus 1 right parenthesis end exponent text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

$text If end text f left square bracket 1 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 1 comma straight infinity right parenthesis text is defined by end text f left parenthesis x right parenthesis equals 2 to the power of x left parenthesis x minus 1 right parenthesis end exponent text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

MathematicsFunctions

$text If end text f colon left curly bracket 1 comma 2 comma 3 comma midline horizontal ellipsis right curly bracket not stretchy rightwards arrow left curly bracket 0 comma plus-or-minus 1 comma plus-or-minus 2 comma midline horizontal ellipsis right curly bracket text is defined by end text f left parenthesis n right parenthesis equals open curly brackets table attributes columnalign center left columnspacing 1em rowspacing 4 pt end attributes row cell n divided by 2 end cell cell text if end text n text is even end text end cell row cell negative open parentheses fraction numerator n minus 1 over denominator 2 end fraction close parentheses end cell cell text if end text n text is odd end text end cell end table close t h e n space f to the power of negative 1 end exponent left parenthesis negative 100 right parenthesis space text is end text$

$text If end text f colon left curly bracket 1 comma 2 comma 3 comma midline horizontal ellipsis right curly bracket not stretchy rightwards arrow left curly bracket 0 comma plus-or-minus 1 comma plus-or-minus 2 comma midline horizontal ellipsis right curly bracket text is defined by end text f left parenthesis n right parenthesis equals open curly brackets table attributes columnalign center left columnspacing 1em rowspacing 4 pt end attributes row cell n divided by 2 end cell cell text if end text n text is even end text end cell row cell negative open parentheses fraction numerator n minus 1 over denominator 2 end fraction close parentheses end cell cell text if end text n text is odd end text end cell end table close t h e n space f to the power of negative 1 end exponent left parenthesis negative 100 right parenthesis space text is end text$

MathematicsFunctions

parallel

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 $f left parenthesis x right parenthesis equals x plus 2 open square brackets x over 5 close square brackets text , then end text$

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 $f left parenthesis x right parenthesis equals x plus 2 open square brackets x over 5 close square brackets text , then end text$

MathematicsFunctions

$text If end text f colon left square bracket 1 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 2 comma straight infinity right parenthesis text is given by end text f left parenthesis x right parenthesis equals x plus 1 over x text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

$text If end text f colon left square bracket 1 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 2 comma straight infinity right parenthesis text is given by end text f left parenthesis x right parenthesis equals x plus 1 over x text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

MathematicsFunctions

$text If end text f colon R minus left curly bracket 5 divided by 2 right curly bracket not stretchy rightwards arrow R minus left curly bracket negative 1 right curly bracket text is defined by end text f left parenthesis x right parenthesis equals fraction numerator 2 x plus 3 over denominator 5 minus 2 x end fraction text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

$text If end text f colon R minus left curly bracket 5 divided by 2 right curly bracket not stretchy rightwards arrow R minus left curly bracket negative 1 right curly bracket text is defined by end text f left parenthesis x right parenthesis equals fraction numerator 2 x plus 3 over denominator 5 minus 2 x end fraction text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

MathematicsFunctions

parallel

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