To determine the range of the function \( g(x) = -3 \sec(2x + 4) - 1 \), we need to take a step-by-step approach:
1. Understand the basic secant function properties:
The secant function, \(\sec(y)\), is defined as \( \sec(y) = \frac{1}{\cos(y)} \). Its range is \( (-\infty, -1] \cup [1, \infty) \).
2. Apply the transformations to \( \sec(y) \) introduced by \( 2x + 4 \) inside the secant function in \( g(x) \):
The expression \( 2x + 4 \) represents a horizontal compression by factor of \(1/2\) and a horizontal shift by \(-4/2\). However, this transformation does not affect the range of \(\sec(y)\).
3. Analyze the transformation applied outside the secant in \( g(x) \) function:
The function \( g(x) = -3 \sec(2x + 4) - 1 \) includes a multiplication by \(-3\) and a vertical shift by \(-1\).
4. Determine the effect on the range:
- Original range of \(\sec(y)\): \((- \infty, -1] \cup [1, \infty)\)
- Multiplication by \(-3\): \(\sec(y) \to -3 \sec(y)\)
This inverts and stretches the range:
[tex]\[ (- \infty, -1] \cup [1, \infty) \xrightarrow{-3 \cdot (\cdot)} (- \infty, -3] \cup [3, \infty) \][/tex]
- Vertical shift by \(-1\): \(-3 \sec(y) \to -3 \sec(y) - 1\)
This translates the range downward by 1 unit:
[tex]\[ (- \infty, -3] - 1 = (- \infty, -4] \][/tex]
[tex]\[ [3, \infty) - 1 = [2, \infty) \][/tex]
5. Combine the intervals:
After applying the transformations, the resulting range for \( g(x) \) is:
[tex]\[
(- \infty, -4] \cup [2, \infty)
\][/tex]
Hence, the range of the function \( g(x) = -3 \sec(2x + 4) - 1 \) is:
[tex]\[
(-\infty,-4] \cup[2, \infty)
\][/tex]
This matches the third option given.
Final Answer:
[tex]\[
(-\infty,-4] \cup[2, \infty)
\][/tex]
