Advice 1: How to determine the range of values

Solve function in daily life not often, but when faced with a need to quickly navigate. Start with the definition of the field values.
How to determine the range of values
Instruction
1
Recall that a function is a dependence of the variable Y from the variable X in which each value of X corresponds to a single value of a variable Y.

The variable X is the independent variable or argument. The variable Y is the dependent variable. It is also considered that the variable Y is a function of the variable X. the function Values equal to the values of the dependent variable.
2
For clarity, write expressions. If the dependent variable Y from variable X is a function, then the abbreviation is written as: y=f(x). (Read: I equals f of x) the Symbol f(x) denote the value function corresponding to the argument value, equal down x.
3
The domain of the function f(x) is called "the set of all valid values of the independent variable x at which the function is defined (has meaning)". Label: D(f) (eng. Define – define.)

Example:
The function f(x) = 1x+1 is defined for all real values of x satisfying x+1 ≠ 0,i.e., x≠-1. Therefore, D(f) = (-∞;-1)U(-1;∞).
4
The range of values of the function y=f(x) is called "the set of all valid values that is independent of the variable y". Symbol: E(f) (eng. Exist – to exist).

Example:
Y=x2 -2x+10; x2 -2x +10 = x2 -2x +1+9+(x-1)2 +9, then the smallest value of the variable y=9 when x=1,so E(y) =[9;∞)
5
All values of the independent variable displays a scope of a function definition. All the values takes the dependent variable, reflect the range of values of the function.
6
The value of the function depends entirely on its scope. In that case, if the scope is not specified, so it changes from minus infinity to plus infinity, therefore the search value of the function at the endpoints of the interval is reduced to the slip limit of this function from minus and plus infinity. Accordingly, if a function set by the formula and its domain is not specified, it is considered that the domain of the function consists of all values of the argument under which the formula makes sense.
7
To find the set of values of the functions need to know the basic properties of elementary functions: scope, values, monotony, continuity, differentiability, even parity, odd parity, periodicity, etc.

Advice 2: How to determine even or odd

Research functions at parity or odd parity is one of the steps of the General algorithm of research of function needed to construct the graph of a function and study its properties. In this step, you define whether the function is even or odd. If the function is not to say that it is even or odd, then we say that is a function of the General form.
How to determine even or odd
Instruction
1
Record function in the form of the dependence y=y(x). For example, y=x+5.
2
Substitute the argument x the argument (-x) and see what happened in the end. Compare with the original function y(x). If y(-x)=y(x) are even function. If y(-x)=-y(x) are odd function. If y(-x) is not equal to y(x) and not equal to -y(x), have the function of the General form.
3
Write the output to this step of the research function. Possible output:y(x) is an even function,y(x) is an odd function,y(x) is a function of the General form.
4
Proceed to the next step of the study of a function using the standard algorithm.
Is the advice useful?
Search