Functions

Definition 1. 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 2. 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 3. 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 ].

Definition 4. The locus of the points with coordinates (х;у) satisfying an equation у=f(x) is called the graph the of function .

Definition 5. A function is even if . A function is odd if .

Definition 6. A function is said to be increasing if larger values of the function correspond to larger values of the argument, i.e., implies .

Definition 7. A function is said to be decreasing if smaller values of the function correspond to larger values of the argument, i.e., implies .

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= cos x, у= sin x, у= tan x, у= cot x, ( trigonometric functions); y= arccos x, у= arcsin x, у= arctan x, у= arccot x (antitrigonometric functions). These functions are studied at school in detail, and do not dwell on them.

Definition 8. A function of the form , where , is called a composite function and denoted by ; it is also the composition of the functions f and j.

Definition 9. A function is said to be periodic with period Т if f(x+T)=f(x) for all х from the domain.

Definition 10. A function is said to be implicit if its equation is given in the form F(x;y)= 0, i.e., у is not explicitly expressed in terms of х.

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