Setting up and recovering functions of several variables

Find the most general continuously differentiable function $g : \mathbb {R} \rightarrow (0,\infty )$ such that the function $h(x,y) = g(x)g(y)$ on $\mathbb {R}^{2}$ is constant on each circle with center $(0,0)$.

Sketch the graph of the function

where

If

find $f\left(1, \dfrac {y}{x}\right)$.

If

find $f(x)$.

Let

If $z=x$ when $y=1$, find the function $f$ and $z$.

Let

If $z=x^{2}$ when $y=0$, find the function $f$ and $z$.

If $f\left(x+y, \dfrac {y}{x}\right)=x^{2}-y^{2}$, find $f(x,y)$.

Find the function $s = f(x ; y)$, if $s$ is the area of a rhombus, $x$ is its perimeter, $y$ is the sum of the lengths of its diagonals. Compute $f(1 ; 2)$.

Find the function $v = f(x ; y)$, if $v$ is the volume of a right circular cone, $x$ is the length of its generatrix, and $y$:

Find the function $v = f(x ; y)$, if $v$ is the volume of a right circular cone, $x$ is the magnitude of the angle between a generatrix and the base plane of the cone, $y$ is the area of the cross-section of the cone by a plane parallel to the base and passing through the center of the ball inscribed in the cone.

Find the function $s = f(x ; y ; z)$, if $s$ is the area of an isosceles trapezoid, $x, y$ are the lengths of the bases, $z$ is the length of the lateral side of the trapezoid. Compute:

1. $f(2 ; 1 ; 2)$;

2. $f(1 ; 4 ; 1)$.

Find the function $s = f(x ; y ; z)$, if $s$ is the area of a triangle, $x, y, z$ are the lengths of its sides.

Find the function $Q = f(x ; y ; z)$, if $Q$ is the lateral surface area of a regular hexagonal frustum, $x, y$ are the sides of the bases, and $z$ is the height of the frustum.

Find the function $v = f\left(x_{1} ; x_{2} ; x_{3} ; x_{4}\right)$, if $v$ is the volume of a tetrahedron, and $x_{i}, i = 1,2,3,4$, are the areas of its faces, with the dihedral angles adjacent to the face of area $x_{1}$ equal to one another. Compute $f(1 ; 1 ; 1 ; 1)$.

Determine whether the equality holds:

Find $f(x ; y)$, if:

Find $f(x ; y) = \sqrt{y}+\varphi (\sqrt{x}-1)$, if $f(x ; 1) = x$.

Find $f(x ; y) = x \varphi (y / x)$, if $f(1 ; y) = \sqrt{1+y^{2}}$.

Find $f(x ; y) = \varphi (x y)+\sqrt{x y} \psi (y / x)$, if $f(1 ; y) = 1, f(x ; x) = x$.

Find $f(x ; y) = \varphi (x y)+\psi (y / x)$, if $f(x ; 1) = \sin (\pi x / 2)$, $f(x ; x) = 1$.

Find $f(x ; y) = \varphi (x+2 \sqrt{y})+\psi (x-2 \sqrt{y}), x \geqslant 0$,

Find $f(x ; y) = \varphi (x)+\psi \left(y+e^{x}\right)$, if $f(0 ; y) = y^{2}$,

Find $u = f(x / z ; y / z-x)$, if $u = x$ when $y = x$.

Find $u = f\left(x^{2} z ; 2 y^{2} z-z^{4}\right)$, if $u = z^{3} / x^{2}$ when $y = x, x > 0$, $z > 0$.

Find $u = f\left(x / z ; x / y-x^{3} y\right)$, if $u = x^{3} z$ when $y = x, y > 0$.

Find $u = f\left(\frac{z-x}{x z} ; \frac{z^{2}}{e^{y}-z}\right)$, if $u = \left(1-\frac{z}{x}\right)^{2}$ when $y = \ln x$, $z > 0$.