In Exercises 53–54, let f(x) = 2 sec x, g(x) = −2 tan x, and h(x) = 2x − π/2. Graph two periods of y = (f∘h)(x).

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Understand the composition of functions: The function \( (f \circ h)(x) \) means \( f(h(x)) \). Given \( f(x) = 2 \sec x \) and \( h(x) = 2x - \frac{\pi}{2} \), substitute \( h(x) \) into \( f \) to get \( (f \circ h)(x) = 2 \sec(2x - \frac{\pi}{2}) \).

Recall the period of the secant function: The basic secant function \( \sec x \) has a period of \( 2\pi \). When the argument is modified to \( 2x - \frac{\pi}{2} \), the period changes according to the coefficient of \( x \).

Calculate the new period: For a function \( \sec(bx + c) \), the period is \( \frac{2\pi}{|b|} \). Here, \( b = 2 \), so the period of \( \sec(2x - \frac{\pi}{2}) \) is \( \frac{2\pi}{2} = \pi \).

Determine the interval for two periods: Since one period is \( \pi \), two periods will span \( 2 \times \pi = 2\pi \). So, the graph of \( y = 2 \sec(2x - \frac{\pi}{2}) \) should be drawn over an interval of length \( 2\pi \) in \( x \).

Identify key points and asymptotes: To graph the function, find where the argument \( 2x - \frac{\pi}{2} \) equals values that cause vertical asymptotes in \( \sec \), such as \( \frac{\pi}{2} + k\pi \), and plot points accordingly to capture the shape over two periods.

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주요 개념

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

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘h)(x) = f(h(x)). Understanding how to substitute h(x) into f(x) is essential for rewriting and analyzing the composite function before graphing.

Secant Function and Its Properties

The secant function, sec x, is the reciprocal of cosine, sec x = 1/cos x. It has vertical asymptotes where cos x = 0 and a period of 2π. Knowing its shape, asymptotes, and period helps in accurately graphing transformations of sec x.

Period of a Composite Trigonometric Function

The period of a composite function like f(h(x)) depends on the inner function h(x). For h(x) = 2x − π/2, the horizontal scaling affects the period of sec x. Calculating the new period is crucial to graphing the function over the correct interval.