Chapter 3. An introduction to calculus

§ 3.1. Functions

Definition. If to each value of an independent variable x one value of a variable y is assigned then у is called a single-valued function of the argument х and is denoted by

.

Definition. The set of values of х for which the value of a function is defined is called the domain (of definition) of this function.

Definition. The set of values of the argument у corresponding to the values х from the domain is called the range of the function.

The domain of a function may be of one of the following types: (1) a closed interval [a;b]; (2) an open interval ]a;b[ , or (a;b); (3) a half-open interval [a;b[ or ]a;b].

Example 1. Find the domain of the function

.

2. Find the domain of the function

3.1.1. The Cartesian coordinate system. Choose a unit length and two intersecting straight lines. The intersection point is taken for the origin 0. If the lines are not perpendicular, then the angle  between them should be known.



0

We have obtained a Cartesian coordinate system.

When , then this Cartesian coordinate system is said to be rectangular.

Definition.The locus of the points with coordinates (х;у) satisfying an equation у=f(x) is called the graph the of function у=f(x).

y

y₁

у_к

х_к 0 х₁ x

Definition. A function у=f(x) is even if

f(–x)=f(x) .

A function is odd if

f(–x)=–f(x) .

y y

у у у

0 x 0 x

–у

The graph of an odd The graph of an even function

function is symmetric is symmetric with respect

With respect to the origin 0. to the 0у axis.

Example. Determine whether the following functions are odd or even:

1. .

We have

f(–x) =; the function is even.

2. . We have

= = ;

the function is odd.

2. we have.

; this function is neither odd nor even; Such functions are called general functions.

Remark. Some students are inclined to believe that all functions are either even or odd; this is not true. General functions are more common than even or odd.

Definition. A function f(x) is said to be increasing if larger values of the function correspond to larger values of the argument, i.e., х₁<x₂ implies f(x₁)<f(x₂).

Definition. A function f(x) is said to be decreasing if smaller values of the function correspond to larger values of the argument, i.e., х₁<x₂ implies f(x₁)>f(x₂).

decreasing y increasing

y₁ y₂

y₂ y₁

x₁ x₂ 0 x₁ x₂ х

3.1.2. Elementary functions. The following functions are considered elementary: y=xⁿ (a power function); у=а^х (an exponential); у=log_аx, у=lgx (a logarithmic function); y=cosx, у=sinx, у=tanx, у=cotx, (trigonometric functions); y=arccosx, у=arcsinx, у=arctanx, у=arccotx (antitrigonometric functions). These functions are studied at school in detail, and do not dwell on them.

The reader is advised to revisit these functions, remember their properties, and construct graphs.