How to determine the monotonicity and parity of a function?

How to judge the monotonicity and parity of a function

Function parity, monotonicity and their identification methods

●Judgement of monotonicity of general functions:

1. Definition method: Suppose x1

2. Derivative method: Derive the differentiable function y=f(x). If y' >0, then y increases monotonically; if y'

●Parity judgment:

1. Definition method: Determine parity by calculating f(-x) to determine whether it is equal to f(x) or -f(x)

2. Use the properties of operations: odd*even=odd odd*odd=even even*even=even odd±odd=odd even±even=even

3. Use derivatives:

The derivative of a differentiable odd function is an even function

The derivative of a differentiable even function is an odd function

●Discrimination of monotonicity of composite functions: If they are the same, they increase, if they are different, they decrease. It means that in F(x)=f(g(x)), if the monotonicity of f and g is the same, then F is an increasing function,

If the monotonicity of f and g is different, then F is a decreasing function.

●Conform to the parity of functions: If one of f and g is an even function, F is an even function. Only when f and g are both odd functions, F is an odd function.

How to identify whether a function is monotonic

The definition of function monotonicity is: If the function y=f(x) is an increasing or decreasing function in a certain interval, then the function y=f(x) is said to have strict monotonicity in this interval.

Note: The monotonicity of a function is also called the increase or decrease of a function

Steps of judgment:

a. Suppose x1 and x2 belong to the given interval, and x1

b. Calculate f(x1)-f(x2) to the simplest

c. Determine the symbol of the above difference

d. Draw a conclusion (if the difference is 0, it is a subtracting function)

Monotonicity is for a certain interval. y=x squared 1 is decreasing on the left side of the coordinate axis and increasing on the right side. It does not strictly increment or decrement

You have to pay attention to a problem. Monotonicity is for a certain interval in the domain of definition. It is a local concept. Some functions are increasing in some intervals in their domain of definition, and in some intervals It’s decreasing

To judge whether a given function is monotonic within its domain, it depends on whether the function is monotonic within the entire domain or a certain interval within a given domain. To put it bluntly, it cannot increase or decrease

Can you understand?

You can see it by drawing the function image

y=x squared 1. This is a quadratic function. Its image is symmetrical about the y-axis. The function is decreasing at (0, negative infinity) and increasing at (0, positive infinity). It is that they have point tonality in these two intervals. But the entire definition domain (negative, positive infinity) cannot be said to be monotonic.

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