# 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 only in a set of real numbers *R*. This means that an argument of a function can adopt only those real values, at which a function is defined, i.e. it also adopts only real values. A set *X* of all admissible real values of an argument *x*, at which a function *y *= *f* ( *x* ) is defined, is called a *domain of * *a function*. A set *Y* of all real values *y*, that a function adopts, is called a *codomain of a function*. Now we can formulate a definition of a function more exactly: *such a rule (law) of a correspondence between a set ***X** and *a set ***Y**, that for each element of a set **X** one and only one element of a set **Y** can be found, is called a function. From this definition it follows, that a function is given if : - the domain of a function *X* is given; - the codomain of a function *Y* is given; - the correspondence rule ( law ), is known. A correspondence rule must be such, that for *each value of an argument only one value* *of a function* can be found. This requirement of a single-valued function is obligatory.
*Monotone function. * If for any two values of an argument * x*_{1} and *x*_{2} from the condition *x*_{2 } >* x*_{1 } it follows *f *(* x*_{2}* *) >* f *(* x*_{1 }), then a function is called *increasing*; if for any *x*_{1} and *x*_{2} from the condition* x*_{2}* *>* x*_{1} it follows *f *(* x*_{2}* *) <* f *(* x*_{1}* *), then a function is called *decreasing*.A function, which only increases or only decreases, is called a *monotone function.*
*Bounded and unbounded functions. * A function is *bounded*, if such positive number *M* exists, that | *f *(* x *) | * ** M * for all values of *x . *If such positive number does not exist, then this function is *unbounded*.
E x a m p l e s. A function, shown on Fig.3, is a bounded, but not monotone function. On Fig.4 quite the opposite, we see a monotone, but unbounded function. ( Explain this, please ! ). *Continuous and discontinuous functions. *A function *y* = *f* ( *x* ) is called a *continuous* *function at a point x* = *a,* if: 1) the function is defined at *x* = *a* , i.e. *f* ( *a* ) exists; 2) a *finite* lim *f* ( *x* ) exists; *x *→ *a * ( see the paragraph "Limits of functions" in the section “Principles of analysis”)
3) *f* ( *a* ) = lim *f* ( *x* ) .
* x* →*a *
If even one from these conditions isn’t executed, this function is called *discontinuous* at the point *x* = *a*. If a function is continuous at *all* points of its domain, it is called a * continuous function*. *Even and odd functions. * If for *any* * x *from a function domain:* f *( – *x* ) = *f* ( *x* ), then this function is called *even*; if *f *( – *x* ) = – *f* ( *x* ), then this function is called * odd *. A graph of an even function is symmetrical relatively *y*-axis ( Fig.5 ), a graph of an odd function is symmetrical relatively the origin of coordinates ( Fig.6 ).
*Periodic function. *A function *f* ( *x* ) is *periodic*, if such *non-zero* number *T* existsthat for *any* *x *from a function domain: *f* ( *x* + *T* ) = *f* ( *x* ). The *least* such number is called a * period of a function*. All trigonometric functions are periodic. E x a m p l e 1 . Prove that sin *x* has a number 2as a period.
S o l u t i o n . We know, that sin ( *x+ *2*n *) = sin *x*, where *n* = 0, ± 1, ± 2, … Hence, adding 2*n * to an argument of a sine doesn’t change its value. Maybe another number with the such property exists ? Assume, that *P* is the such number, i.e. the equality:
sin ( *x + **P * ) = sin *x*, is valid for any value of *x*. Then this is valid for *x* = / 2 , i.e.
sin (* * / 2 *+ * * P* ) = sin / 2 = 1. But sin (* * / 2 *+ ** P * ) = cos *P* according to the reduction formula.Then from the two last expressions it follows, that cos *P* = 1, but we know, that this equality is right only if *P* = 2*n*. Because the least non-zero number of 2*n * is 2, this is a period of sin *x*. It is proved analogously, that 2 is also a period for cos *x* . Prove, please, that functions tan *x* and cot *x* have as a period.E x a m p l e 2. What number is a period for the function sin 2*x* ?
S o l u t i o n . Consider
sin 2*x* = sin ( 2*x + *2*n * ) = sin [ 2 ( *x* + *n * ) ]. We see, that adding *n * to an argument* x*, doesn’t change the function value. The least non-zero number of *n *is , so this is a period of sin 2*x *.*Zeros of function. *An argument value, at which a function is equal to zero, is called a *zero ( root ) of the function. *It can be that a function has some zeros.* *For instance, the function *y* = *x* ( *x* + 1 ) ( *x* – 3) has the three zeros: *x *= 0,* x *= – 1, * x* = 3 . Geometrically, a zero of a function is *x*-coordinate of a point of intersection of the function graph and *x*-axis. On Fig.7 a graph of a function with zeros *x *= *a , x* = *b* and * x * =* c *is represented.
*Asymptote. *If a graph of a function unboundedly approaches to some straight line at itstaking off an origin of coordinates, then this straight line is called an *asymptote*.
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