Question

Как найти промежутки и область значений любой функции? Скажите пожайлуста очень. Завтра соревнования.

Answer 1

Область значений - это все значения графика по координате "x"

Answer 2

Область значений это минимальное и максимальное значение у
Промежуток знакопостоянства это когда у больше нуля, а когда меньше
по графику определяются проще
Промежуток знакопостоянства:
(У>0) те значения Х, при которых у выше оси ОХ
И наоборот (у<0), где ниже ОХ
Область значений от самой низкой до самой высокой точки
Промежутки знакопостоянства всегда отмечаются по оси ОХ
Т.е. значение-х
Область значений-по оси ОУ
Следовательно, значение - у

Answer 3

Finding Intervals and Range of a Function

To find the intervals and range of any function, you can follow these steps:

1. Determine the Domain of the Function: - The domain of a function is the set of all possible input values (x-values) for the function. It determines where the function is defined. - To find the domain, look for any restrictions on the function, such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

2. Identify Critical Points: - Critical points are the points where the derivative of the function is zero or undefined. These points can help determine the intervals of increase and decrease for the function.

3. Test Intervals for Increasing and Decreasing Behavior: - Use the first derivative test to determine where the function is increasing or decreasing. This will help identify the intervals of the function.

4. Find the Range of the Function: - Once you have identified the intervals of the function, you can evaluate the function at the endpoints of these intervals to find the range.

5. Consider Special Cases: - For some functions, there may be special cases to consider, such as asymptotes or discontinuities, which can affect the intervals and range of the function.

Example:

Let's consider the function f(x) = x^2 - 4x + 4.

1. Domain of the Function: - The domain of a quadratic function is all real numbers, so there are no restrictions on the domain.

2. Critical Points: - To find the critical points, we take the derivative of the function: f'(x) = 2x - 4. - Setting f'(x) = 0 gives us x = 2 as the critical point.

3. Intervals of Increase and Decrease: - We can use the first derivative test to determine that the function is decreasing for x < 2 and increasing for x > 2.

4. Range of the Function: - Evaluating the function at the endpoints of the intervals gives us the range: f(2) = 0, and as x approaches positive or negative infinity, f(x) approaches positive infinity.

By following these steps, you can find the intervals and range of any function. Good luck with your competition tomorrow!