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Mathematics

Functions

Easy

Question

$text If end text f colon N not stretchy rightwards arrow Z space f left parenthesis n right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1em end attributes row cell fraction numerator n minus 1 over denominator 2 end fraction comma text when end text n text is odd end text end cell row cell negative n over 2 comma text when end text n text is even end text end cell end table close$ Identify the type of function

One -one and onto
Not one - one but onto
One - one but not onto
Neither one - one nor onto

The correct answer is: One -one and onto

$text When end text n text is even, let end text f open parentheses 2 m subscript 1 close parentheses equals f open parentheses 2 m subscript 2 close parentheses table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell text or end text space of 1em minus fraction numerator 2 m subscript 1 over denominator 2 end fraction equals negative fraction numerator 2 m subscript 2 over denominator 2 end fraction end cell row cell text or end text space of 1em m subscript 1 equals m subscript 2 end cell end table text When end text n text is odd, let end text f open parentheses 2 m subscript 1 plus 1 close parentheses equals f open parentheses 2 m subscript 2 plus 1 close parentheses text or end text space of 1em fraction numerator 2 m subscript 1 plus 1 minus 1 over denominator 2 end fraction equals fraction numerator 2 m subscript 2 plus 1 minus 1 over denominator 2 end fraction text or end text m subscript 1 equals m subscript 2 text Therefore, end text f left parenthesis x right parenthesis text is one-one. end text text Also, when end text n text is even, end text minus n over 2 equals negative fraction numerator 2 m over denominator 2 end fraction equals negative m text When end text n text is odd, end text fraction numerator n minus 1 over denominator 2 end fraction equals fraction numerator 2 m plus 1 minus 1 over denominator 2 end fraction equals m H e n c e comma space t h e space r a n g e space o f space t h e space f u n c t i o n space i s space Z. space T h e r e f o r e comma space f u n c t i o n space i s space o n t o.$

Related Questions to study

Which of the following functions from Z to it self are bijections?

Which of the following functions from Z to it self are bijections?

MathematicsFunctions

Let f(x) be a function and k be a positive real number such that
$f left parenthesis x plus k right parenthesis plus f left parenthesis x right parenthesis equals 0 space f o r space a l l space x element of R. space t h e n space f i n d space t h e space p e r i o d space o f space f left parenthesis x right parenthesis space$

Let f(x) be a function and k be a positive real number such that
$f left parenthesis x plus k right parenthesis plus f left parenthesis x right parenthesis equals 0 space f o r space a l l space x element of R. space t h e n space f i n d space t h e space p e r i o d space o f space f left parenthesis x right parenthesis space$

MathematicsFunctions

If f(x) is polynomial function satisfying $text end text f left parenthesis x right parenthesis times f open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f open parentheses 1 over x close parentheses text and end text text end text f left parenthesis 4 right parenthesis equals 65 comma text then find end text f left parenthesis 6 right parenthesis$

If f(x) is polynomial function satisfying $text end text f left parenthesis x right parenthesis times f open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f open parentheses 1 over x close parentheses text and end text text end text f left parenthesis 4 right parenthesis equals 65 comma text then find end text f left parenthesis 6 right parenthesis$

MathematicsFunctions

parallel

Let f be a function satisfying f(xy) = $fraction numerator f left parenthesis x right parenthesis over denominator y end fraction$ for all positive real numbers x and y if f(30) =20, then find the value of f(40)

Let f be a function satisfying f(xy) = $fraction numerator f left parenthesis x right parenthesis over denominator y end fraction$ for all positive real numbers x and y if f(30) =20, then find the value of f(40)

MathematicsFunctions

$text Let a function end text f left parenthesis x right parenthesis text satisfies end text f left parenthesis x right parenthesis plus f left parenthesis 2 x right parenthesis plus f left parenthesis 2 minus x right parenthesis plus f left parenthesis 1 plus x right parenthesis equals x straight for all x element of R.$ Then find the value of f(0)

$text Let a function end text f left parenthesis x right parenthesis text satisfies end text f left parenthesis x right parenthesis plus f left parenthesis 2 x right parenthesis plus f left parenthesis 2 minus x right parenthesis plus f left parenthesis 1 plus x right parenthesis equals x straight for all x element of R.$ Then find the value of f(0)

MathematicsFunctions

The function f(x) is defined for all real x. If f(a+b) =f(ab) $for all a text and end text b$ $text and end text f open parentheses negative 1 half close parentheses equals negative 1 half$ then find the value of f(1005)

The function f(x) is defined for all real x. If f(a+b) =f(ab) $for all a text and end text b$ $text and end text f open parentheses negative 1 half close parentheses equals negative 1 half$ then find the value of f(1005)

MathematicsFunctions

parallel

$text Let end text f left parenthesis x right parenthesis equals x plus f left parenthesis x minus 1 right parenthesis text for end text straight for all x element of R. text If end text f left parenthesis 0 right parenthesis equals 1 comma text find end text f left parenthesis 100 right parenthesis$

$text Let end text f left parenthesis x right parenthesis equals x plus f left parenthesis x minus 1 right parenthesis text for end text straight for all x element of R. text If end text f left parenthesis 0 right parenthesis equals 1 comma text find end text f left parenthesis 100 right parenthesis$

MathematicsFunctions

$text If end text f subscript n left parenthesis x right parenthesis equals open parentheses tan x over 2 close parentheses left parenthesis 1 plus sec x right parenthesis left parenthesis 1 plus sec 2 x right parenthesis midline horizontal ellipsis open parentheses 1 plus sec 2 to the power of n x close parentheses text end text$
Then which of the following is not true?

$text If end text f subscript n left parenthesis x right parenthesis equals open parentheses tan x over 2 close parentheses left parenthesis 1 plus sec x right parenthesis left parenthesis 1 plus sec 2 x right parenthesis midline horizontal ellipsis open parentheses 1 plus sec 2 to the power of n x close parentheses text end text$
Then which of the following is not true?

MathematicsFunctions

$text Let end text sum from k equals 1 to 10 of f left parenthesis a plus k right parenthesis equals 16 open parentheses 2 to the power of 10 minus 1 close parentheses text where the function end text straight f text satisfies end text f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis text end text$
For all natural numbers x,y and f(1) =2. Then the natural number 'a' is

$text Let end text sum from k equals 1 to 10 of f left parenthesis a plus k right parenthesis equals 16 open parentheses 2 to the power of 10 minus 1 close parentheses text where the function end text straight f text satisfies end text f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis text end text$
For all natural numbers x,y and f(1) =2. Then the natural number 'a' is

MathematicsFunctions

parallel

$text If end text straight f left parenthesis straight x right parenthesis text is a function such that end text f left parenthesis x y right parenthesis equals f left parenthesis x right parenthesis plus f left parenthesis y right parenthesis text and end text f left parenthesis 2 right parenthesis equals 1 text then end text straight f left parenthesis straight x right parenthesis equals$

$text If end text straight f left parenthesis straight x right parenthesis text is a function such that end text f left parenthesis x y right parenthesis equals f left parenthesis x right parenthesis plus f left parenthesis y right parenthesis text and end text f left parenthesis 2 right parenthesis equals 1 text then end text straight f left parenthesis straight x right parenthesis equals$

MathematicsFunctions

$text If end text straight f left parenthesis straight x right parenthesis text is a function such that end text f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis text and end text f left parenthesis 3 right parenthesis equals 125 text then end text straight f left parenthesis straight x right parenthesis equals$

$text If end text straight f left parenthesis straight x right parenthesis text is a function such that end text f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis text and end text f left parenthesis 3 right parenthesis equals 125 text then end text straight f left parenthesis straight x right parenthesis equals$

MathematicsFunctions

If f(x) is a polynomial function such that $f left parenthesis x right parenthesis f open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f open parentheses 1 over x close parentheses text and end text f left parenthesis 3 right parenthesis equals negative 80$ then f(x)

If f(x) is a polynomial function such that $f left parenthesis x right parenthesis f open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f open parentheses 1 over x close parentheses text and end text f left parenthesis 3 right parenthesis equals negative 80$ then f(x)

MathematicsFunctions

parallel

$text If end text f left parenthesis x right parenthesis text is a polynomial function satisfying end text f left parenthesis x right parenthesis times f open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f open parentheses 1 over x close parentheses text and end text f left parenthesis 4 right parenthesis equals 257 comma text then end text f left parenthesis 3 right parenthesis equals$

$text If end text f left parenthesis x right parenthesis text is a polynomial function satisfying end text f left parenthesis x right parenthesis times f open parentheses 1 over x close parentheses equals f left parenthesis x right parenthesis plus f open parentheses 1 over x close parentheses text and end text f left parenthesis 4 right parenthesis equals 257 comma text then end text f left parenthesis 3 right parenthesis equals$

MathematicsFunctions

The set of all values of a for which the function $f colon R rightwards arrow R text given by end text$
$f left parenthesis x right parenthesis equals x cubed plus left parenthesis a plus 2 right parenthesis x squared plus 3 a x plus 5 text is one-one is end text$

The set of all values of a for which the function $f colon R rightwards arrow R text given by end text$
$f left parenthesis x right parenthesis equals x cubed plus left parenthesis a plus 2 right parenthesis x squared plus 3 a x plus 5 text is one-one is end text$

MathematicsFunctions

Let f: (-1,1) → B, be a function defined by $f left parenthesis x right parenthesis equals t an to the power of negative 1 end exponent fraction numerator 2 x over denominator 1 minus x squared end fraction$ then f is both one-one and onto when B is the interval

Let f: (-1,1) → B, be a function defined by $f left parenthesis x right parenthesis equals t an to the power of negative 1 end exponent fraction numerator 2 x over denominator 1 minus x squared end fraction$ then f is both one-one and onto when B is the interval

MathematicsFunctions

parallel

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