The Concept of a Function

Find the domain of the following function: $y=\dfrac {x^{2}}{1+x}$.

Find the domain of the following function: $y=\sqrt{3x-x^{3}}$.

Find the domain of the following function: $y=(x-2)\sqrt{\dfrac {1+x}{1-x}}$.

Find the domain of the following functions:

Find the domain of the following function: $y=\sqrt{\sin (\sqrt{x})}$.

Find the domain of the following function: $y=\sqrt{\cos x^{2}}$.

Find the domain of the following function: $y=\log \left(\sin \dfrac {\pi }{x}\right)$.

Find the domain of the following function: $y=\dfrac {\sqrt{x}}{\sin \pi x}$.

Find the domain of the following function: $y=\arcsin \dfrac {2x}{1+x}$.

Find the domain of the following function: $y=\arccos (2\sin x)$.

Find the domain of the following function: $y=\lg [\cos (\lg x)]$.

Find the domain of the following function: $y=(x-\left|x\right|)\sqrt{-\sin^{2}\pi x}$.

Find the domain of the following function: $y=\operatorname {ctg}\pi x+\arccos (2^{x})$.

Find the domain of the following function: $y=\arcsin (1-x)+\lg (\lg x)$.

Find the domain of the following function: $y=(2x)!$.

Find the domain and range of the following function: $y=\sqrt{2+x-x^{2}}$.

Find the domain and range of the following function: $y=\lg (1-2\cos x)$.

Find the domain and range of the following function: $y=\arccos \dfrac {2x}{1+x^{2}}$.

Find the domain and range of the following function: $y=\arcsin \left(\lg \dfrac {x}{10}\right)$.

Find the domain and range of the following function: $y=(-1)^{x}$.

In a triangle $ABC$ with base $AC=b$ and height $BD=h$ (Fig.1.2), a rectangle $KLMN$ with height $NM=x$ is inscribed. Express the perimeter $P$ of rectangle $KLMN$ and its area $S$ as functions of $x$.

Plot the graphs of the functions $P=P(x)$ and $S=S(x)$.

In a triangle $ABC$, side $AB=6$ cm, side $AC=8$ cm, angle $BAC=x$. Express the side $BC=a$ and the area $ABC=S$ as functions of the variable $x$. Plot the graphs of the functions $a=a(x)$ and $S=S(x)$.

In an isosceles trapezoid $ABCD$ (Fig.1.7), with bases $AD=a,BC=b (a>b)$ and height $HB=h$, draw a line $MN\parallel BH$, where $MN$ is at distance $AM=x$ from vertex $A$. Express the area $S$ of the figure $ABNMA$ as a function of the variable $x$. Plot the graph of the function $S=S(x)$.

On the closed interval $0\leqslant x\leqslant 1$ of the $Ox$ axis a mass of 2 grams is uniformly distributed, while at the two points $x=2$ and $x=3$ on this axis there are concentrated masses of 1 gram each.

Let $m(x)$ be the value of the mass lying in the interval $(-\infty ,x)$. Find the analytic expression for the function $m=m(x) (-\infty <x<+\infty )$, and plot the graph of this function.

The function $y=\operatorname {sgn}x$ is defined by the following method:

Plot the graph of this function. Prove that

The function $y=[x]$ (the integer part of the number $x$) is defined as follows: if $x=n+r$, where $n$ is an integer and $0\leqslant r<1$, then

Plot the graph of this function.

Let:

denote the number of primes not exceeding the number $x$. For values of the independent variable in $0\leqslant x\leqslant 20$, plot the graph of this function.

Onto which set $E_{y}$ does the function $y=f(x)$ map the set $E_{x}$, if: $y=x^{2},E_{x}=\left\{ 1\leqslant x\leqslant 2\right\}$.

Onto which set $E_{y}$ does the function $y=f(x)$ map the set $E_{x}$, if: $y=\lg x,E_{x}=\left\{ 10<x<1000\right\}$.

Onto which set $E_{y}$ does the function $y=f(x)$ map the set $E_{x}$, if: $y=\dfrac {1}{\pi }\operatorname {arcctg}x,E_{x}=\left\{ -\infty <x<+\infty \right\}$.

Onto which set $E_{y}$ does the function $y=f(x)$ map the set $E_{x}$, if: $y=\operatorname {ctg}\dfrac {\pi x}{4},E_{x}=\left\{ 0<\left|x\right|\leqslant 1\right\}$.

Onto which set $E_{y}$ does the function $y=f(x)$ map the set $E_{x}$, if: $y=\left|x\right|,E_{x}=\left\{ 1\leqslant \left|x\right|\leqslant 2\right\}$.

The variable $x$ runs through the interval $0<x<1$; what set does the variable $y$ run through, if: $y=a+(b-a)x$.

The variable $x$ runs through the interval $0<x<1$; what set does the variable $y$ run through, if: $y=\dfrac {1}{1-x}$.

The variable $x$ runs through the interval $0<x<1$; what set does the variable $y$ run through, if: $y=\dfrac {x}{2x-1}$.

The variable $x$ runs through the interval $0<x<1$; what set does the variable $y$ run through, if: $y=\sqrt{x-x^{2}}$.

The variable $x$ runs through the interval $0<x<1$; what set does the variable $y$ run through, if: $y=\operatorname {ctg}\pi x$.

The variable $x$ runs through the interval $0<x<1$; what set does the variable $y$ run through, if: $y=x+[2x]$.

Let:

find $f(0),f(1),f(2),f(3),f(4)$.

Let:

find $f(-1),f(-0.001),f(100)$.

Let:

find $f(0.9),f(0.99),f(0.999),f(1)$.

Let:

find $f(-2),f(-1),f(0),f(1),f(2)$.

Let:

find $f(0),f(-x),f(x+1),f(x)+1,f\left(\dfrac {1}{x}\right),\dfrac {1}{f(x)}$.

Let:

Let:

Let:

prove that $f(x+3)-3f(x+2)+3f(x+1)-f(x)\equiv 0$.

If $f(0)=-2,f(3)=5$, find the linear integer function:

What are $f(1)$ and $f(2)$ (linear interpolation)?

If $f(-2)=0,f(0)=1,f(1)=5$. Find the quadratic integer rational function:

What are $f(-1)$ and $f(0.5)$ (quadratic interpolation)?

Let $f(-1)=0,f(0)=2,f(1)=-3,f(2)=5$. Find the cubic integer rational function:

Let $f(0)=15,f(2)=30,f(4)=90$, find the function of the form

Prove: for the linear function

if the values of the independent variable $x=x_{n} (n=1,2,\ldots )$ form an arithmetic sequence, then the corresponding function values $y_{n}=f(x_{n}) (n=1,2,\ldots )$ also form an arithmetic sequence.

Prove: for the exponential function

if the values of the independent variable $x=x_{n} (n=1,2,\ldots )$ form an arithmetic sequence, then the corresponding function values $y_{n}=f(x_{n}) (n=1,2,\ldots )$ form a geometric sequence.

Suppose the function $f(u)$ is defined for $0<u<1$, find the domain of each of the following functions:

Let:

Prove: $f(x+y)+f(x-y)=2f(x)\cdot f(y)$.

Let:

Find $z$, if:

Find $\varphi [\varphi (x)],\psi [\psi (x)],\varphi [\psi (x)]$ and $\psi [\varphi (x)]$, given: $\varphi (x)=x^{2}$ and $\psi (x)=2^{x}$.

Find $\varphi [\varphi (x)],\psi [\psi (x)],\varphi [\psi (x)]$ and $\psi [\varphi (x)]$, given: $\varphi (x)=\operatorname {sgn}x$ and $\psi (x)=\dfrac {1}{x}$.

Find $\varphi [\varphi (x)],\psi [\psi (x)],\varphi [\psi (x)]$ and $\psi [\varphi (x)]$, given:

Let:

find $f[f(x)],f\left\{ f[f(x)]\right\}$.

Let:

If $f(x)=\dfrac {x}{\sqrt{1+x^{2}}}$, find $f_{n}(x)$.

Let:

find $f(x)$.

Let:

find $f(x)$.

Let:

find $f(x)$.

Prove that the following function is monotonically increasing on the given interval: $f(x)=x^{2} (0\leqslant x<+\infty )$.

Prove that the following function is monotonically increasing on the given interval: $f(x)=\sin x \left(-\dfrac {\pi }{2}\leqslant x\leqslant \dfrac {\pi }{2}\right)$.

Prove that the following function is monotonically increasing on the given interval: $f(x)=\operatorname {tg}x \left(-\dfrac {\pi }{2}<x<\dfrac {\pi }{2}\right)$.

Prove that the following function is monotonically increasing on the given interval: $f(x)=2x+\sin x (-\infty <x<+\infty )$.

Prove that the following function is monotonically decreasing on the given interval: $f(x)=x^{2} (-\infty <x\leqslant 0)$.

Prove that the following function is monotonically decreasing on the given interval: $f(x)=\cos x (0\leqslant x\leqslant \pi )$.

Prove that the following function is monotonically decreasing on the given interval: $f(x)=\operatorname {ctg}x (0<x<\pi )$.

Investigate the monotonicity of the following functions:

Can an inequality be taken term-by-term to logarithms?

Let $\varphi (x),\psi (x)$ and $f(x)$ be monotonically increasing functions. Prove: if

then

Find the inverse function $x=\varphi (y)$ and its domain of existence, if: $y=2x+3 (-\infty <x<+\infty )$.

Find the inverse function $x=\varphi (y)$ and its domain of existence, if: $y=x^{2}$.

Find the inverse function $x=\varphi (y)$ and its domain of existence, if: $y=\dfrac {1-x}{1+x} (x\neq -1)$.

Find the inverse function $x=\varphi (y)$ and its domain of existence, if: $y=\sqrt{1-x^{2}}$.

Find the inverse function $x=\varphi (y)$ and its domain of existence, if: $y=\operatorname {sh}x$, where $\operatorname {sh}x=\dfrac {1}{2}(e^{x}-e^{-x}) (-\infty <x<+\infty )$.

Find the inverse function $x=\varphi (y)$ and its domain of existence, if: $y=\operatorname {th}x$, where $\operatorname {th}x=\dfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}} (-\infty <x<+\infty )$.

Find the inverse function $x=\varphi (y)$ and its domain of existence, if:

A function $f(x)$ is defined on a symmetric interval $(-l,l)$, and if

then $f(x)$ is called an even function; if

then $f(x)$ is called an odd function.

Determine which of the following given functions are even and which are odd:

Prove that any function $f(x)$ defined on a symmetric interval $(-l,l)$ can be expressed as the sum of an even function and an odd function.

If there exists a number $T>0$ (the period of the function — in the broad sense) such that for all considered independent variables $x$ the equality

holds, then the function $f(x)$ is called a periodic function.

Determine which of the following given functions are periodic functions, and find their smallest periods. Let:

Prove: for the Dirichlet function

any rational number is its period.

Prove that the sum and product of two periodic functions defined on a common set, whose periods are commensurable, are also periodic functions.

Prove: if for the function $f(x) (-\infty <x<+\infty )$ the equality

(where $k$ and $T$ are positive constants) holds, then

(where $a$ is a constant greater than zero, and $\varphi (x)$ is a function with period $T$).