Instruction

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The properties of evenness and unevenness of the function is determined based on the influence of the sign of the argument to its value. This influence is shown in the graph of the functions with some symmetry. In other words, a following property of the parity if f(-x) = f(x), i.e. the sign of the argument does not affect the value of the function, and the odd, if true equality f(-x) = -f(x).

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Odd function graphically looks symmetric about the point of intersection of coordinate axes, even relative to the y-axis. An example of an even function can be a parabola x2, odd – f = x3.

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Example No. 1Исследовать parity

*function*x2/(4·x2 - 1).Solution:Substitute in the given function x instead of x. You will see that the sign functions do not change, since the argument in both cases is present in even degree, which neutralizes the negative sign. Therefore, the investigated function is even.4

Example No. 2Проверить function is even or odd: f = -x2 + 5·x.Solution:As in the previous example, substitute –x instead of x: f(-x) = -x2 – 5·x. It is obvious that f(x) ≠ f(-x) and f(-x) ≠ -f(x) therefore the function has the properties of neither parity nor unevenness. Such a function is called indifferent, or function of the General form.

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To investigate the function of even parity and odd parity can also obvious when plotting or finding domain function. In the first example the scope is a set x ∈ (-∞; 1/2) ∪ (1/2; +∞). The graph of the function is symmetrical relative to Oy axis, so the function is even.

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Mathematics is the first study of properties of elementary functions, and then the obtained knowledge is transferred to the study of more complex functions. Basic are power functions with integer exponent, exponential form a^x where a>0, logarithmic, and trigonometric functions.

# 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.Instruction

1

Record function in the form of the dependence y=y(x). For example, y=x+5.

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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.

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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.

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Proceed to the next step of the study of a function using the standard algorithm.

# Advice 3: How to investigate the function of parity

Examination of functions of even parity and odd parity helps to build the graph of the function and to investigate the nature of its behavior. For this study it is necessary to compare the function written for the argument "x" and argument "-x".

Instruction

1

Record function, the study on which it is to be in the form y=y(x).

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Replace the function argument with "-x". Substitute the argument in the function expression.

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Simplify the expression.

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So you got the same function written for the arguments "x" and "-x". Look at these two entries.

If y(-x)=y(x), it is an even function.

If y(-x)=-y(x), then it is an odd function.

If about the function, we cannot say that y(-x)=y(x) or y(-x)=-y(x), the parity property is a function of the General form. That is, it is neither even nor odd.

If y(-x)=y(x), it is an even function.

If y(-x)=-y(x), then it is an odd function.

If about the function, we cannot say that y(-x)=y(x) or y(-x)=-y(x), the parity property is a function of the General form. That is, it is neither even nor odd.

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Write down your insights. Now you can use them in plotting functions or further analytical study of the properties of the function.

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To talk about parity and odd functions even in the case where the already set schedule function. For example, a graph was the result of physical experiment.

If the graph of a function symmetric about the y-axis, then y(x) is an even function.

If the graph of a function symmetric about the x-axis, x(y) is an even function. x(y) is the inverse of the function y(x).

If the graph of a function symmetric about the origin (0,0), then y(x) is an odd function. Odd is also the inverse function x(y).

If the graph of a function symmetric about the y-axis, then y(x) is an even function.

If the graph of a function symmetric about the x-axis, x(y) is an even function. x(y) is the inverse of the function y(x).

If the graph of a function symmetric about the origin (0,0), then y(x) is an odd function. Odd is also the inverse function x(y).

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It is important to remember that the concept of parity and of odd functions has a direct relationship with the domain of the function. If, for example, is even or odd the function exists at x=5, then it does not exist and when x=-5, what can be said about the function of the General form. When establishing parity and odd, pay attention to the domain of the function.

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Examination of functions of even parity and odd parity correlated with the presence of many function values. To find the set of values of even functions, it is sufficient to consider half of the functions to the right or to the left of zero. If x>0 an even function y(x) takes values from A to b, then the same values it will take and when x<0.

To find the set of values taken by an odd function, is also sufficient to consider only one part of the function. If x>0 odd function y(x) takes a range of values from A to b, when x<0 it will make a symmetrical range of values from (- ) to (-A).

To find the set of values taken by an odd function, is also sufficient to consider only one part of the function. If x>0 odd function y(x) takes a range of values from A to b, when x<0 it will make a symmetrical range of values from (- ) to (-A).