Construction of real functions

Construction of real functions . In mathematical analysis , mainly (even in complex analysis), from the elementary functions, by means of rational operations and composition of functions, other new real functions can be constructed, which will increase the wide collection of real functions. If in the same domain X, a subset of the set R of all real numbers , two real functions f and h are defined, they can be defined with them, new functions using rational operations, just as they are done in number systems.

Summary

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• 1 Employing rational operations
• 2 Function and a number
• 3 Other derived functions
• 4 Composition of functions
• 5 See also
• 6 References
• 7 Source

Employing rational operations

Sum of functions

the function f + h by (f + h) (x) = f (x) + h (x), where x is in X

Function difference

the function fh by (f – h) (x) = f (x) – h (x), in which x belongs to X

Product of functions

the function f × h by (f × h) (x) = f (x) × h (x), in which x is in X

The quotient function

the function f / h by f / h (x) = f (x) / h (x), here x is a member of X ^[1]

The inverse of a function

the function f ^-1 by f ^-1 (x) = {1} / {f (x)}, where x in X

Function and a number

If f is a function defined in the set X, part of R, set of all real numbers, then for arbitrary numbers a, b and c, each ≠ 0, the following functions can be defined from f

1. f (x) + b which means vertical displacementof ordinate. Let f (x) = x ² ; at f “(x) = x ² +3, all the ordinate goes up to 3
2. cf (x) ordinate dilationor compression . Let Let f (x) = x ² ; at f “(x) = 0.5x ² ; each ordinate is compressed; the graph is” widened “.
3. f (x – a) horizontal displacementof abscissa. Being f (x) = x ² ; At f “(x) = (x-3) ² ; (0,0) changes to (3,0). Identical copy.
4. f (cx) dilationor compression of abscissa ^[2]

Other derived functions

1. -f (x); makes the -f graph symmetric about the Ox axis.
2. f (-x); it acts in the sense that the graph of f (-x) comes out symmetric from f with respect to the axis Oy.
3. f (| x |); involves the symmetric points of the points to the right of the Oy axis.
4. | f (x) |; negative ordinates become positive ordinates

Composition of functions

Given the function f of domain A, codomain B and the function h whose domain is part of B, its codomain is C. We are going to define the composition of functions denoted

f º h, by means of the formula (f º h) (x) = f [h (x)] being its domain, part of A and its codomain, part C.