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Maths-

General

Easy

Question

Let $f colon open square brackets 1 half comma 1 close square brackets not stretchy rightwards arrow left square bracket negative 1 comma 1 right square bracket$ is given by $f left parenthesis x right parenthesis equals 4 x cubed minus 3 x text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis text is given by end text$

$\cos [\frac{1}{3} \cos^{- 1} x]$
$3 \cos (\sin^{- 1} x)$
$3 \sin^{- 1} (\cos x)$
$\sin (\frac{1}{3} \cos^{- 1} x)$

hint Hint:

In this question, we have to find inverse of function f(x) $f colon open square brackets 1 half comma 1 close square brackets not stretchy rightwards arrow left square bracket negative 1 comma 1 right square bracket$ . Firstly, we will assume x to be cos $theta$ and y to be cos 3 $theta$ . Then, we will find f(x) for x = cos $theta$ and later find the value of $theta$ from that equation. Now, finding inverse of f(y). Further, replacing y with x to get the required inverse of f(x).

The correct answer is: $cos invisible function application open square brackets 1 third cos to the power of negative 1 end exponent invisible function application x close square brackets$

$f left parenthesis x right parenthesis equals 4 x cubed minus 3 x w h e r e comma space f open square brackets 1 half comma 1 close square brackets rightwards arrow open square brackets negative 1 comma 1 close square brackets L e t space x equals space cos space theta space a n d space y equals cos space 3 theta f left parenthesis cos space theta right parenthesis equals 4 cos cubed theta minus 3 space cos space theta space space space space space space space space space space space space space equals space cos space 3 space theta a s space 3 theta space equals cos to the power of negative 1 end exponent y space space space space space space space space space space theta equals 1 third cos to the power of negative 1 end exponent y f to the power of negative 1 end exponent left parenthesis cos space 3 theta right parenthesis equals cos space theta a s space y space equals space f left parenthesis x right parenthesis rightwards double arrow f to the power of negative 1 end exponent left parenthesis y right parenthesis equals x f to the power of negative 1 end exponent left parenthesis cos space 3 theta right parenthesis space equals space cos space theta f to the power of negative 1 end exponent left parenthesis y right parenthesis space equals space cos space open parentheses 1 third cos to the power of negative 1 end exponent y close parentheses f to the power of negative 1 end exponent left parenthesis x right parenthesis equals cos open parentheses 1 third cos to the power of negative 1 end exponent x close parentheses$

Related Questions to study

If $f left parenthesis x right parenthesis equals fraction numerator 10 to the power of x minus 10 to the power of negative x end exponent over denominator 10 to the power of x plus 10 to the power of negative x end exponent end fraction text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

If $f left parenthesis x right parenthesis equals fraction numerator 10 to the power of x minus 10 to the power of negative x end exponent over denominator 10 to the power of x plus 10 to the power of negative x end exponent end fraction text then end text f to the power of negative 1 end exponent left parenthesis x right parenthesis equals$

If $f left parenthesis x right parenthesis equals fraction numerator e to the power of x plus e to the power of negative x end exponent over denominator 2 end fraction$ ,then the inverse function of $f left parenthesis x right parenthesis text is end text$

If $f left parenthesis x right parenthesis equals fraction numerator e to the power of x plus e to the power of negative x end exponent over denominator 2 end fraction$ ,then the inverse function of $f left parenthesis x right parenthesis text is end text$

If $f left parenthesis x right parenthesis equals x squared plus x comma x equals 10 comma delta x equals 0.1$ observe the following
$text l) end text delta f equals 2.11$
$text II) end text d f equals 2.1$
$text III) Relative error in end text x text is end text 1$
The true statements are

If $f left parenthesis x right parenthesis equals x squared plus x comma x equals 10 comma delta x equals 0.1$ observe the following
$text l) end text delta f equals 2.11$
$text II) end text d f equals 2.1$
$text III) Relative error in end text x text is end text 1$
The true statements are

parallel

Statement I : $f colon A not stretchy rightwards arrow B text is one - one and end text g colon B not stretchy rightwards arrow C text is a one-one function, then gof end text colon A not stretchy rightwards arrow C$ is one - one
Statement II : $text If end text f colon A not stretchy rightwards arrow B comma g colon B not stretchy rightwards arrow A$ are two functions such that $g o f equals I subscript A text and end text fog equals I subscript B comma text then end text f equals$ $g to the power of negative 1 end exponent$
Staatement III : $f left parenthesis x right parenthesis equals sec squared invisible function application x minus tan squared invisible function application x g left parenthesis x right parenthesis equals cosec squared invisible function application x minus cot squared invisible function application x comma text then end text f equals g$
Which of the above statement/s is/are true.

Statement I : $f colon A not stretchy rightwards arrow B text is one - one and end text g colon B not stretchy rightwards arrow C text is a one-one function, then gof end text colon A not stretchy rightwards arrow C$ is one - one
Statement II : $text If end text f colon A not stretchy rightwards arrow B comma g colon B not stretchy rightwards arrow A$ are two functions such that $g o f equals I subscript A text and end text fog equals I subscript B comma text then end text f equals$ $g to the power of negative 1 end exponent$
Staatement III : $f left parenthesis x right parenthesis equals sec squared invisible function application x minus tan squared invisible function application x g left parenthesis x right parenthesis equals cosec squared invisible function application x minus cot squared invisible function application x comma text then end text f equals g$
Which of the above statement/s is/are true.

A glass capillary sealed at the upper end is of length 0.11 m and internal diameter $2 cross times 1 0 to the power of negative 5 end exponent$ m. The tube is immersed vertically into a liquid of surface tension $5.06 cross times 1 0 to the power of negative 2 end exponent$ N/m. To what length has the capillary to be immersed so that the liquid level inside and outside the capillary becomes the same?

A glass capillary sealed at the upper end is of length 0.11 m and internal diameter $2 cross times 1 0 to the power of negative 5 end exponent$ m. The tube is immersed vertically into a liquid of surface tension $5.06 cross times 1 0 to the power of negative 2 end exponent$ N/m. To what length has the capillary to be immersed so that the liquid level inside and outside the capillary becomes the same?

physics-General

Water flows through a frictionless duct with a cross-section varying as shown in fig. . Pressure p at points along the axis is represented by: A resume water to be non-viscour

Water flows through a frictionless duct with a cross-section varying as shown in fig. . Pressure p at points along the axis is represented by: A resume water to be non-viscour

physics-General

parallel

A cubical block of wood 10 cm on a side floats at the interface between oil and water with its lower surface horizontal and 4 cm below the interface. The density of oil is $0.6 g c m to the power of negative 3 end exponent$ . The mass of block is

A cubical block of wood 10 cm on a side floats at the interface between oil and water with its lower surface horizontal and 4 cm below the interface. The density of oil is $0.6 g c m to the power of negative 3 end exponent$ . The mass of block is

physics-General

Water flows through a frictionless duct with a cross-section varying as shown in fig. Pressure p at points along the axis is represented by

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physics-General

The diagram shows a cup of tea seen from above. The tea has been stirred and is now rotating without turbulence. A graph showing the speed v with which the liquid is crossing points at a distance X from O along a radius XO would look like

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$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals log subscript e invisible function application x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 2 log subscript e invisible function application x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals log subscript e invisible function application x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 2 log subscript e invisible function application x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals 2 to the power of x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 3 to the power of x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

$f colon R to the power of plus not stretchy rightwards arrow R text defined by end text f left parenthesis x right parenthesis equals 2 to the power of x comma x element of left parenthesis 0 comma 1 right parenthesis comma f left parenthesis x right parenthesis equals 3 to the power of x comma x element of left square bracket 1 comma straight infinity right parenthesis text is end text$

A small spherical solid ball is dropped from a great height in a viscous liquid. Its journey in the liquid is best described in the diagram given below by the

A small spherical solid ball is dropped from a great height in a viscous liquid. Its journey in the liquid is best described in the diagram given below by the

physics-General

parallel

Two communicating vessels contain mercury. The diameter of one vessel is n times larger than the diameter of the other. A column of water of height h is poured into the left vessel. The mercury level will rise in the right-hand vessel (s = relative density of mercury and $rho$ = density of water) by

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There are two identical small holes of area of cross-section a on the opposite sides of a tank containing a liquid of density $rho$ . The difference in height between the holes is h. Tank is resting on a smooth horizontal surface. Horizontal force which will has to be applied on the tank to keep it in equilibrium is

There are two identical small holes of area of cross-section a on the opposite sides of a tank containing a liquid of density $rho$ . The difference in height between the holes is h. Tank is resting on a smooth horizontal surface. Horizontal force which will has to be applied on the tank to keep it in equilibrium is

physics-General

A cylinder containing water up to a height of 25 cm has a hole of cross-section $fraction numerator 1 over denominator 4 end fraction c m to the power of 2 end exponent$ in its bottom. It is counterpoised in a balance. What is the initial change in the balancing weight when water begins to flow out

A cylinder containing water up to a height of 25 cm has a hole of cross-section $fraction numerator 1 over denominator 4 end fraction c m to the power of 2 end exponent$ in its bottom. It is counterpoised in a balance. What is the initial change in the balancing weight when water begins to flow out

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parallel

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