Given \( f(x)=\sec x \) and \( g(x)=\sin x \), what is the composed function \( f(g(x)) \) and what are its domain and range? (1 point) \( f(g(x))=\sec (\sin x) \) domain: all real numbers, \( x \neq n \pi \), where \( n \) is an integer range: \( 1 \leq x \leq 1.85 \) \( f(g(x))=\sin (\sec x) \) domain: all real numbers, \( x \neq n \pi \), where \( n \) is an integer range: \( -1 \leq x \leq 1 \) \( f(g(x))=\sec (\sin x) \) domain: all real numbers range: \( 1 \leq x \leq 1.85 \) \( f(g(x))=\sin (\sec x) \) domain all real numbers range: \( -1 \leq x \leq 1 \)

To find the composed function \( f(g(x)) \) where \( f(x) = \sec x \) and \( g(x) = \sin x \), we will follow these steps: 1. **Compose the functions**: \[ f(g(x)) = f(\sin x) = \sec(\sin x) \] 2. **Determine the domain of \( f(g(x)) \)**: - The function \( \sec x \) is defined for all \( x \) except where \( \cos x = 0 \). This occurs at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \). - Since \( g(x) = \sin x \) outputs values in the range \([-1, 1]\), we need to check where \( \sec(\sin x) \) is defined. The cosine function is non-zero for all values in the interval \([-1, 1]\). - Therefore, the domain of \( f(g(x)) = \sec(\sin x) \) is all real numbers, as \( \sin x \) will never output values that make \( \sec(\sin x) \) undefined. 3. **Determine the range of \( f(g(x)) \)**: - The range of \( \sin x \) is \([-1, 1]\). - The secant function \( \sec y = \frac{1}{\cos y} \) has a range of \( (-\infty, -1] \cup [1, \infty) \) for \( y \) values outside of \( \frac{\pi}{2} + n\pi \). - Since \( \sin x \) only takes values in \([-1, 1]\), we need to evaluate \( \sec(y) \) for \( y \) in this interval: - At \( y = -1 \): \( \sec(-1) = \sec(-\frac{\pi}{2}) \) which is defined. - At \( y = 1 \): \( \sec(1) = \sec(\frac{\pi}{2}) \) which is also defined. - The minimum value of \( \sec(y) \) occurs at \( y = 0 \) where \( \sec(0) = 1 \). - The maximum value occurs at the endpoints \( y = -1 \) and \( y = 1 \), which are both greater than or equal to 1. Thus, the composed function is: \[ f(g(x)) = \sec(\sin x) \] - **Domain**: All real numbers. - **Range**: \( [1, \sec(1)] \) where \( \sec(1) \) is approximately \( 1.8508 \). Now, let's summarize the correct answer: - \( f(g(x)) = \sec(\sin x) \) - Domain: All real numbers - Range: \( [1, \sec(1)] \) or approximately \( [1, 1.8508] \) The correct option is: \( f(g(x))=\sec (\sin x) \) domain: all real numbers range: \( 1 \leq x \leq 1.85 \)