Show that the function f colon R rightwards arrow left curly bracket x element of R space colon space minus 1 less than x less than 1 right curly bracket defined by f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 plus vertical line x vertical line end fraction is one-one and onto.

Asked by abhinavsaini123 | 1st Jun, 2015, 09:41: PM

Expert Answer:

size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis size 14px space size 14px equals size 14px space size 14px y size 14px equals fraction numerator size 14px x over denominator size 14px 1 size 14px plus size 14px x end fraction size 14px rightwards double arrow size 14px x size 14px equals fraction numerator size 14px y over denominator size 14px 1 size 14px minus size 14px y end fraction size 14px therefore size 14px space size 14px x size 14px greater or equal than size 14px 0 size 14px space size 14px f size 14px o size 14px r size 14px space size 14px a size 14px l size 14px l size 14px space size 14px y size 14px element of size 14px left square bracket size 14px 0 size 14px comma size 14px space size 14px 1 size 14px right parenthesis size 14px. size 14px T size 14px h size 14px u size 14px s size 14px comma size 14px space size 14px f size 14px i size 14px s size 14px space size 14px a size 14px n size 14px space size 14px o size 14px n size 14px t size 14px o size 14px space size 14px f size 14px u size 14px n size 14px c size 14px t size 14px i size 14px o size 14px n size 14px space size 14px f size 14px r size 14px o size 14px m size 14px space size 14px left square bracket size 14px 0 size 14px comma size 14px space size 14px infinity size 14px right parenthesis size 14px space size 14px t size 14px o size 14px space size 14px left square bracket size 14px 0 size 14px comma size 14px space size 14px 1 size 14px right parenthesis size 14px.  size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis size 14px equals fraction numerator size 14px x over denominator size 14px 1 size 14px plus size 14px vertical line size 14px x size 14px vertical line end fraction size 14px equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator size 14px x over denominator size 14px 1 size 14px plus size 14px x end fraction size 14px space size 14px i size 14px f size 14px 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14px 1 size 14px plus size 14px x end fraction  size 14px L size 14px e size 14px t size 14px space size 14px x size 14px comma size 14px space size 14px y size 14px element of size 14px R size 14px space size 14px s size 14px u size 14px c size 14px h size 14px space size 14px t size 14px h size 14px a size 14px t size 14px space size 14px x size 14px comma size 14px y size 14px greater or equal than size 14px 0 size 14px. size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis size 14px space size 14px equals size 14px space size 14px f size 14px left parenthesis size 14px y size 14px right parenthesis size 14px rightwards double arrow fraction numerator size 14px x over denominator size 14px 1 size 14px plus size 14px x end fraction size 14px equals fraction numerator size 14px y over denominator size 14px 1 size 14px plus size 14px y end fraction size 14px rightwards double arrow size 14px x size 14px plus size 14px x size 14px y size 14px equals size 14px y 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parenthesis size 14px space size 14px b size 14px e size 14px space size 14px a size 14px n size 14px y size 14px space size 14px r size 14px e size 14px a size 14px l size 14px space size 14px n size 14px u size 14px m size 14px b size 14px e size 14px r size 14px space size 14px t size 14px h size 14px e size 14px n size 14px comma size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis size 14px space size 14px equals size 14px space size 14px y size 14px rightwards double arrow fraction numerator size 14px x over denominator size 14px 1 size 14px plus size 14px x end fraction size 14px equals size 14px y size 14px rightwards double arrow size 14px x size 14px equals fraction numerator size 14px y over denominator size 14px 1 size 14px minus size 14px y end fraction   size 14px therefore size 14px space size 14px x size 14px greater or equal than size 14px 0 size 14px space size 14px f size 14px o size 14px r size 14px space size 14px a size 14px l size 14px l 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size 14px a size 14px l size 14px l size 14px space size 14px y size 14px element of size 14px left parenthesis size 14px minus size 14px 1 size 14px comma size 14px space size 14px 0 size 14px right parenthesis size 14px space  size 14px therefore size 14px f size 14px space size 14px i size 14px s size 14px space size 14px o size 14px n size 14px t size 14px o size 14px space size 14px f size 14px r size 14px o size 14px m size 14px space size 14px left parenthesis size 14px minus size 14px infinity size 14px comma size 14px space size 14px 0 size 14px right parenthesis size 14px space size 14px t size 14px o size 14px space size 14px left parenthesis size 14px minus size 14px 1 size 14px comma size 14px space size 14px 0 size 14px right parenthesis   size 14px H size 14px e size 14px n size 14px c size 14px e size 14px space size 14px p size 14px r size 14px o size 14px v size 14px e size 14px d

Answered by Vijaykumar Wani | 2nd Jun, 2015, 11:08: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.