In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = tan(2x - π)

검증된 단계별 안내

Step 1: Identify the function type. The given function is y = tan(2x - π), which is a transformation of the basic tangent function y = tan(x).

Step 2: Determine the domain of the function. The tangent function is undefined where its argument is an odd multiple of π/2. Set the argument 2x - π equal to (2n+1)π/2, where n is an integer, to find the values of x that make the function undefined.

Step 3: Solve for x in the equation 2x - π = (2n+1)π/2. This will give you the values of x that need to be excluded from the domain. Rearrange the equation to find x = (π/2)(2n+1)/2 + π/2.

Step 4: Express the domain. The domain of y = tan(2x - π) is all real numbers except x = (π/2)(2n+1)/2 + π/2, where n is an integer.

Step 5: Determine the range of the function. The range of the tangent function is all real numbers, so the range of y = tan(2x - π) is also all real numbers.

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주요 개념

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the tangent function, it is important to identify values that would make the function undefined, such as where the argument of the tangent function equals π/2 + kπ, where k is any integer. In this case, the domain of y = tan(2x - π) can be determined by solving for x in the equation 2x - π = π/2 + kπ.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the tangent function, the range is all real numbers, as it can take any value from negative to positive infinity. Therefore, regardless of the specific transformations applied to the tangent function, such as the horizontal shift in y = tan(2x - π), the range remains the same.

Transformation of Functions

Transformations of functions involve shifting, stretching, or compressing the graph of a function. In the case of y = tan(2x - π), the '2' indicates a horizontal compression by a factor of 1/2, while the '-π' indicates a horizontal shift to the right by π/2. Understanding these transformations is crucial for accurately determining the domain and range of the function.

In Exercises 19–32, find the (a) domain and (b) range.𝔂 = tan(2x - π)