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Mathematics

Functions

Easy

Question

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)

$- 1 / 2$
$- 3 / 2$
$- 5 / 2$
$- 7 / 2$

The correct answer is: $negative 1 divided by 2$

$text We have end text f left parenthesis a plus b right parenthesis equals f left parenthesis a b right parenthesis straight for all a text and end text b text Put end text space of 1em a equals 0 therefore space of 1em f left parenthesis b right parenthesis equals f left parenthesis 0 right parenthesis equals k left parenthesis text say end text right parenthesis text Putting end text b equals 0 text in given relation, end text f left parenthesis a right parenthesis equals f left parenthesis 0 right parenthesis equals k text Thus end text f left parenthesis a right parenthesis equals f left parenthesis b right parenthesis equals k straight for all a comma b text Hence, end text f left parenthesis x right parenthesis text is a constant function. end text table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell therefore space of 1em f left parenthesis 1005 right parenthesis equals f open parentheses negative 1 half close parentheses end cell row cell equals negative 1 half end cell end table$

Related Questions to study

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

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

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

$text The function end text f colon left square bracket 0 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 0 comma straight infinity right parenthesis text defined by end text f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 plus x end fraction text is end text$

$text The function end text f colon left square bracket 0 comma straight infinity right parenthesis not stretchy rightwards arrow left square bracket 0 comma straight infinity right parenthesis text defined by end text f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 plus x end fraction text is end text$

MathematicsFunctions

Let A = { x ε R : x is not a positive integer }. Define a function f : A → R as $f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x minus 1 end fraction text then end text f text is end text$

Let A = { x ε R : x is not a positive integer }. Define a function f : A → R as $f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x minus 1 end fraction text then end text f text is end text$

MathematicsFunctions

$text If end text f colon N not stretchy rightwards arrow bold italic R text is defined by end text f left parenthesis 1 right parenthesis equals negative 1 text and end text f left parenthesis n plus 1 right parenthesis equals 3 f left parenthesis n right parenthesis plus 2 text for end text n greater than 1 comma text then end text f text is end text$

$text If end text f colon N not stretchy rightwards arrow bold italic R text is defined by end text f left parenthesis 1 right parenthesis equals negative 1 text and end text f left parenthesis n plus 1 right parenthesis equals 3 f left parenthesis n right parenthesis plus 2 text for end text n greater than 1 comma text then end text f text is end text$

MathematicsFunctions

parallel

$text Define end text f colon Z not stretchy rightwards arrow Z text by end text f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row cell x divided by 2 end cell cell left parenthesis x text is even end text right parenthesis end cell row 0 cell left parenthesis x text is odd end text right parenthesis end cell end table close text then end text f text is end text$

$text Define end text f colon Z not stretchy rightwards arrow Z text by end text f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign center center columnspacing 1em rowspacing 4 pt end attributes row cell x divided by 2 end cell cell left parenthesis x text is even end text right parenthesis end cell row 0 cell left parenthesis x text is odd end text right parenthesis end cell end table close text then end text f text is end text$

MathematicsFunctions

$text If end text f colon R not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals x comma text if end text x greater than 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 5 x minus 2 comma text if end text x less or equal than 2 comma space space text then end text f text is end text space$

$text If end text f colon R not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals x comma text if end text x greater than 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 5 x minus 2 comma text if end text x less or equal than 2 comma space space text then end text f text is end text space$

MathematicsFunctions

For real x, $text let end text f left parenthesis x right parenthesis equals x cubed plus 5 x plus 1 comma text end text$ then

For real x, $text let end text f left parenthesis x right parenthesis equals x cubed plus 5 x plus 1 comma text end text$ then

MathematicsFunctions

parallel

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