State the domain and range of the functions. y=1-2 sec((1)/(2) x+π)

Name: Problem 62, Chapter 6: Algebra and Trigonometry
Uploaded: 2021-11-07T22:36:10-08:00
Duration: 1 min 26 s
Channel: James Kiss
Description: State the domain and range of the functions. y=1-2 sec((1)/(2) x+π)

step 1 I want to state the domain and range of the function, y equals 1 minus 2 secant times 1 1 half x

State the domain and range of the functions. y=1-2...

Key Concepts

Range Transformations in Trigonometry

The inherent range of the basic secant function is (-?, -1] U [1, ?). When vertical transformations are applied, such as scaling by a factor or shifting, the range is modified accordingly. Understanding how these transformations affect the original range is essential for accurately determining the final set of values that the function can take.

Determining Domain of Trigonometric Functions

For functions that involve a secant, the domain must exclude the x-values that result in a cosine value of zero, as this would lead to division by zero. The process involves solving the equation derived from the transformation inside the cosine function where cosine equals zero, and then determining the intervals where the function is defined.

Secant Function

The secant function is defined as the reciprocal of the cosine function. Key properties include its periodicity and the fact that it becomes undefined when the cosine is zero. Understanding the secant function is crucial in trigonometry for analyzing graphs and determining restrictions in the domain resulting from these undefined points.

Trigonometric Function Transformations

Transformations involve changing the function by applying horizontal shifts, vertical shifts, stretches, or reflections. For a function of the form y = a sec(bx + c) + d, these transformations affect both the position and the shape of the graph. The horizontal transformation, influenced by the factor b and the constant c, adjusts where the function's periods begin and end, while the vertical transformation, determined by a and d, scales and shifts the basic range of the secant function.

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