Graphical solving of inequalities Approximate solution of inequalities. Graphical solving of inequalities in one unknown. Graphical solving of systems of simultaneous inequalities in two unknowns. Intersection of solutions. Graphical representation of functions permits to solve approximately inequalities in one unknown and systems of simultaneous inequalities in one and two unknowns. To solve an inequality […]

## Graphical solving of equations

Graphical solving of equations Approximate solution of equations. Graphical solving of equations in one unknown. Graphical solving of systems of simultaneous equations in two unknowns. Graphical representation of functions permits to solve approximately any equation in one unknown and a system of two simultaneous equations in two unknowns. To solve a system of two simultaneous […]

## Elementary functions and their graphs

Elementary functions and their graphs Proportional values. Linear function. Inverse proportionality. Hyperbola. Quadratic function. Quadratic parabola. Power function. Cubic parabola. Exponential function. Logarithmic function. Trigonometric functions. Sinusoid. Intervals of monotony. Inverse trigonometric functions. 1. Proportional values. If variables y and x are direct proportional, then the functionaldependence between them is represented by the equation: […]

## Composite function

Composite function Consider the function: y = sin 2 ( 2x ) . Actually, this record means the following chain of functional transformations: u = 2x –> v = sin u –> y = v 2 , that can be written by symbols of functional dependences in a general view as: u = f1 ( […]

## Inverse function

Inverse function If to change an argument and a function by roles, then x will be a function of y. In this case we say about a new function, called an inverse function. Assume, we have a function: v = u 2 , where u is an argument and v is a function. If to […]

## Basic notions and properties of functions

Basic notions and properties of functions Function. Domain and codomain of a function. Rule (law) of correspondence. Monotone function. Bounded and unbounded function. Continuous and discontinuous function. Even and odd function. Periodic function. Period of a function. Zeros (roots) of a function. Asymptote. Domain and codomain of function. In elementary mathematics we study functions […]

## Coordinates. Graphical representation of functions

Coordinates. Graphical representation of functions Coordinates. Coordinate system. Cartesian coordinates. Axes of coordinates. Origin of coordinates. Abscissa and ordinate. Graphical representation of functions. Graph of a functional dependence. Coordinates. Two mutually perpendicular straight lines XX’ and YY’ ( Fig.1 ) form a coordinate system, called Cartesian coordinates. Straight lines XX’ and YY’ are called axes of […]

## Designation of functions

Designation of functions Let y be some function of variable x; moreover, it is not essential, how this function is given: by formula or by table or by any other way. Only the fact of existence of this functional dependence is important. This fact is written as: y = f ( x ). The letter […]

## Representation of function by formula and table

Representation of function by formula and table Many of functions can be represented ( exactly or approximately ) by simple formulas. For example, the dependence between an area S of a circle and its radius r is given by the formula S = r 2 ; the previous example shows the dependence between a height h […]

## Functional dependence between two variables

Functional dependence between two variables Functional dependence. Argument ( independent variable ). Function. Single-valued function. Multiple-valued function. Two variables x and y are tied by a functional dependence, if for each value of one of them it is possible to receive by the certain rule one or some values of another. E x a m […]