Math Problems Vault

Problem 759

Find the inverse function of the linear fractional function

y=\dfrac {ax+b}{cx+d} \quad (ad-bc\neq 0)

Under what circumstances is the inverse function the same as the given function?

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Problem 760

Let

y=x+\left[x\right] ,

find the inverse function $x=x(y)$ .

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Problem 761

Prove that there is a unique continuous function $y=y(x)$ ( $-\infty <x<+\infty$ ) satisfying Kepler's equation

y-\varepsilon \sin y=x \quad (0\leqslant \varepsilon <1) .

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Problem 762

Prove that the equation

\operatorname {ctg}x=kx

has, for every real number $k$ ( $-\infty <k<+\infty$ ), a unique continuous root $x=x(k)$ in the interval $0<x<\pi$ .

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Problem 763

Can a non-monotone function $y=f(x)$ ( $-\infty <x<+\infty$ ) have a single-valued inverse function?

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Problem 764

Under what circumstances are the function $y=f(x)$ and the inverse function $x=f^{-1}(y)$ the same function?

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Problem 765

Prove that the inverse function of the discontinuous function

y=(1+x^{2})\operatorname {sgn}x

is continuous.

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Problem 766

Prove that if the function $f(x)$ is defined and strictly monotone on the closed interval $\left[a,b\right]$ , and

\lim _{n\to \infty }f(x_{n})=f(a) \quad (a\leqslant x_{n}\leqslant b) ,

then

\lim _{n\to \infty }x_{n}=a .

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Problem 767

Find the continuous single-valued branch of the inverse function of $y=x^{2}$ .

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Problem 768

Find the continuous single-valued branch of the inverse function of $y=2x-x^{2}$ .

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Problem 769

Find the continuous single-valued branch of the inverse function of $y=\dfrac {2x}{1+x^{2}}$ .

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Problem 770

Find the continuous single-valued branch of the inverse function of $y=\sin x$ .

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Problem 771

Find the continuous single-valued branch of the inverse function of $y=\cos x$ .

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Problem 772

Find the continuous single-valued branch of the inverse function of $y=\operatorname {tg}x$ .

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Problem 773

Prove that the set of values of the continuous function $y=1+\sin x$ corresponding to the interval $0<x<2\pi$ is a line segment.

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Problem 774

Prove the identity

\arcsin x+\arccos x=\dfrac {\pi }{2} .

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Problem 775

Prove the identity

\operatorname {arctg}x+\operatorname {arctg}\dfrac {1}{x}=\dfrac {\pi }{2}\operatorname {sgn}x \quad (x\neq 0) .

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Problem 776

Prove the theorem for adding arctangents:

\operatorname {arctg}x+\operatorname {arctg}y=\operatorname {arctg}\dfrac {x+y}{1-xy}+\varepsilon \pi ,

where $\varepsilon =\varepsilon (x,y)$ is a function taking one of the three values $0,1,-1$ .

When the value of $x$ is fixed, for which values of $y$ can the function $\varepsilon$ be discontinuous? On the $Oxy$ plane, sketch the corresponding regions where the function $\varepsilon$ is continuous, and find the value of this function on each of the regions obtained.

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Problem 777

Prove the theorem for adding arcsines:

\arcsin x+\arcsin y=(-1)^{\varepsilon }\arcsin \left(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right)+\varepsilon \pi \quad (\left|x\right|\leqslant 1,\left|y\right|\leqslant 1) ,

where, if $xy\leqslant 0$ or $x^{2}+y^{2}\leqslant 1$ , $\varepsilon =0$ ; if $xy>0$ and $x^{2}+y^{2}>1$ , $\varepsilon =\operatorname {sgn}x$ .

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Problem 778

Prove the theorem for adding arccosines:

\arccos x+\arccos y=(-1)^{\varepsilon }\arccos \left(xy-\sqrt{1-x^{2}}\sqrt{1-y^{2}}\right)+2\varepsilon \pi \quad (\left|x\right|\leqslant 1,\left|y\right|\leqslant 1) ,

where, if $x+y\geqslant 0$ , $\varepsilon =0$ ; if $x+y<0$ , $\varepsilon =1$ .

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Problem 779

(a)(b)

Sketch the graph of the function:

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(a)

$y=\arcsin x-\arcsin \sqrt{1-x^{2}}$ ;

(b)

$y=\arcsin \left(2x\sqrt{1-x^{2}}\right)-2\arcsin x$ .

Problem 780

The function $y=y(x)$ is given by the equations:

x=\operatorname {arctg}t , \quad y=\operatorname {arcctg}t \quad (-\infty <t<+\infty ) ,

find this function. On what domain is this function defined?

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Problem 781

Let

x=\operatorname {ch}t , \quad y=\operatorname {sh}t \quad (-\infty <t<+\infty ) .

Over what domain of variation of the parameter $t$ can the variable $y$ be regarded as a single-valued function of the variable $x$ ? Find the expression for $y$ on each such domain.

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Problem 782

What is the necessary and sufficient condition for the system of equations

x=\varphi (t) , \quad y=\phi (t)

to define $y$ as a single-valued function of $x$ ?

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Problem 783

Under what conditions do the two systems of equations

x=\varphi (t) , \quad y=\phi (t) \quad (a<t<b)

and

x=\varphi \left[\chi (\tau )\right] , \quad y=\phi \left[\chi (\tau )\right] \quad (\alpha <\tau <\beta )

define the same function $y=y(x)$ ?

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Problem 784

Let the functions $\varphi (x)$ and $\phi (x)$ be defined and continuous on the interval $(a,b)$ , and let

A=\inf _{a<x<b}\varphi (x) , \quad B=\sup _{a<x<b}\varphi (x) .

Under what circumstances does there exist a single-valued function $f(x)$ defined on the interval $(A,B)$ such that, for $a<x<b$ , $\phi (x)=f\left[\varphi (x)\right]$ ?

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