Graph y = sin x and y = ⌊sin x⌋ together. What are the domain and range of ⌊sin x⌋?

검증된 단계별 안내

Step 1: Understand the function y = sin x. The sine function is periodic with a period of 2π, and it oscillates between -1 and 1. Its domain is all real numbers, and its range is [-1, 1].

Step 2: Understand the function y = ⌊sin x⌋. The floor function, denoted by ⌊x⌋, returns the greatest integer less than or equal to x. For y = ⌊sin x⌋, this means it will take the value of the greatest integer less than or equal to the sine of x.

Step 3: Determine the domain of y = ⌊sin x⌋. Since the sine function is defined for all real numbers, the domain of ⌊sin x⌋ is also all real numbers.

Step 4: Determine the range of y = ⌊sin x⌋. Since sin x oscillates between -1 and 1, ⌊sin x⌋ will take integer values within this range. Specifically, ⌊sin x⌋ can be -1, 0, or 1 depending on the value of sin x.

Step 5: Graph y = sin x and y = ⌊sin x⌋ together. On the graph, y = sin x will be a smooth wave oscillating between -1 and 1, while y = ⌊sin x⌋ will be a step function that jumps between -1, 0, and 1 at the points where sin x crosses integer values.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sine Function

The sine function, denoted as y = sin x, is a periodic function that oscillates between -1 and 1. It is defined for all real numbers and is commonly used in trigonometry. The graph of y = sin x exhibits a wave-like pattern, repeating every 2π radians, which is essential for understanding its behavior and properties.

Floor Function

The floor function, denoted as ⌊x⌋, maps a real number x to the largest integer less than or equal to x. When applied to the sine function, ⌊sin x⌋ takes the output of sin x and rounds it down to the nearest integer. This transformation alters the continuous wave of sin x into a step function, which has distinct characteristics in terms of its domain and range.

Domain and Range

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). For the floor function applied to sin x, the domain remains all real numbers, but the range becomes limited to the integers -1 and 0, reflecting the integer outputs of the floor function when applied to the sine values.