Functions defined parametrically

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Problem 780 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.8 "Inverse Functions. Functions Given Parametrically"

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?

Problem 781 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.8 "Inverse Functions. Functions Given Parametrically"

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.

Problem 782 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.8 "Inverse Functions. Functions Given Parametrically"

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 $x$ ?

Problem 783 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.8 "Inverse Functions. Functions Given Parametrically"

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 $y=y(x)$ ?

Problem 784 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.8 "Inverse Functions. Functions Given Parametrically"

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 $a<x<b$ , $\phi (x)=f\left[\varphi (x)\right]$ ?

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.