Consider the function g(x) = -2 csc (4x + π). What is the domain of g? What is its range?

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Identify the function: \( g(x) = -2 \csc(4x + \pi) \). The cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function, \( \sin(\theta) \).

Determine the domain: The sine function is undefined where it equals zero, so \( \csc(\theta) \) is undefined where \( \sin(\theta) = 0 \). Solve \( \sin(4x + \pi) = 0 \) to find the values of \( x \) that make \( g(x) \) undefined.

Solve for \( x \) in \( \sin(4x + \pi) = 0 \): The sine function is zero at integer multiples of \( \pi \), so set \( 4x + \pi = n\pi \), where \( n \) is an integer. Solve for \( x \) to find the excluded values.

Determine the range: The range of \( \csc(\theta) \) is \(( -\infty, -1 ] \cup [ 1, \infty )\). Since \( g(x) = -2 \csc(4x + \pi) \), multiply the range of \( \csc(\theta) \) by \(-2\) to find the range of \( g(x) \).

Conclude the domain and range: The domain excludes the values of \( x \) found in step 3, and the range is the result from step 4.

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주요 개념

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Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important to note that the cosecant function is undefined wherever the sine function is zero, which occurs at integer multiples of π. Understanding the behavior of the cosecant function is crucial for determining the domain and range of functions that involve it.

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 function g(x) = -2 csc(4x + π), the domain is restricted by the values that make the sine function zero. Identifying these values allows us to determine where the function g(x) is undefined, thus establishing its domain.

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 cosecant function, the range is limited to values outside the interval (-1, 1) because csc(x) approaches infinity as sin(x) approaches zero. For g(x) = -2 csc(4x + π), the range can be derived from the properties of the cosecant function, taking into account the vertical stretch and reflection caused by the coefficient -2.