Question

Given f left parenthesis x right parenthesis equals secant x and g left-parenthesis x right-parenthesis equals sine x, what is the composed function f left parenthesis g left parenthesis x right parenthesis right parenthesis and what are its domain and range?
(1 point)
Responses

f left parenthesis g left parenthesis x right parenthesis right parenthesis equals secant left parenthesis sine x right parenthesis
domain: all real numbers, x not equals n pi, where n is an integer
range: 1 less than or equals x less than or equals 1.85
Image with alt text: f left parenthesis g left parenthesis x right parenthesis right parenthesis equals secant left parenthesis sine x right parenthesis domain: all real numbers, Image with alt text: x not equals n pi , where n is an integer range: Image with alt text: 1 less than or equals x less than or equals 1.85

f left parenthesis g left parenthesis x right parenthesis right parenthesis equals sine left parenthesis secant x right parenthesis
domain: all real numbers, x not equals n pi, where n is an integer
range: negative 1 less than or equals x less than or equals 1
Image with alt text: f left parenthesis g left parenthesis x right parenthesis right parenthesis equals sine left parenthesis secant x right parenthesis domain: all real numbers, Image with alt text: x not equals n pi , where n is an integer range: Image with alt text: negative 1 less than or equals x less than or equals 1

f left parenthesis g left parenthesis x right parenthesis right parenthesis equals secant left parenthesis sine x right parenthesis
domain: all real numbers
range: 1 less than or equals x less than or equals 1.85
Image with alt text: f left parenthesis g left parenthesis x right parenthesis right parenthesis equals secant left parenthesis sine x right parenthesis domain: all real numbers range: Image with alt text: 1 less than or equals x less than or equals 1.85

f left parenthesis g left parenthesis x right parenthesis right parenthesis equals sine left parenthesis secant x right parenthesis
domain: all real numbers
range: negative 1 less than or equals x less than or equals 1
Image with alt text: f left parenthesis g left parenthesis x right parenthesis right parenthesis equals sine left parenthesis secant x right parenthesis domain: all real numbers range: Image with alt text: negative 1 less than or equals x less than or equals 1

Answer 1

To find the composed function \( f(g(x)) \) where \( f(x) = \sec(x) \) and \( g(x) = \sin(x) \), we evaluate:

\[
f(g(x)) = f(\sin(x)) = \sec(\sin(x)) = \frac{1}{\cos(\sin(x))}
\]

### Domain of \( f(g(x)) \)

1. The function \( \sec(x) \) (and hence \( \sec(\sin(x)) \)) is undefined whenever \( \cos(\sin(x)) = 0 \).
2. The cosine function is zero at odd multiples of \( \frac{\pi}{2} \). We need the values of \( \sin(x) \) that lead to these points.
3. Specifically, \( \sin(x) = \frac{\pi}{2} + n\pi \) for \( n \in \mathbb{Z} \).

However, the range of \( \sin(x) \) is limited to \([-1, 1]\). The values where \( \cos(y) = 0 \) within this range are \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \), which are not reached by \( \sin(x) \). Therefore, within the real numbers, \( \sec(\sin(x)) \) is defined for all real \( x \).

### Range of \( f(g(x)) \)

1. The output \( \sec(\sin(x)) \) takes values based on the input \( \sin(x) \), which varies from \(-1\) to \(1\).
2. \( \sec(y) \) has a range of \( (-\infty, -1] \cup [1, \infty) \) when \( y \in [-1, 1] \).
3. Since \( \sin(x) \) cannot produce the value where \( \sec(y) \) is between \(-1\) and \(1\), the range of \( f(g(x)) \) will be:
\[
(-\infty, -1] \cup [1, \infty)
\]

### Conclusion

So, we can summarize:

- The composed function:
\[
f(g(x)) = \sec(\sin(x))
\]
- Domain:
\[
\text{All real numbers}
\]
- Range:
\[
(-\infty, -1] \cup [1, \infty)
\]