To determine the domain of the piecewise function [tex]\( f(x) \)[/tex], we need to examine the intervals on which each piece of the function is defined. The function [tex]\( f(x) \)[/tex] is specified as follows:
[tex]\[
f(x) = \begin{cases}
4x + 2 & \text{if } x \leq -5 \\
-\frac{3}{2} x + 3 & \text{if } -2 < x < 4 \\
-\frac{1}{2} x + 7 & \text{if } 5 \leq x
\end{cases}
\][/tex]
We will analyze each case to gather all intervals where [tex]\( f(x) \)[/tex] is defined:
1. For [tex]\( x \leq -5 \)[/tex]:
- The function [tex]\( f(x) = 4x + 2 \)[/tex] is defined for all values of [tex]\( x \)[/tex] that are less than or equal to [tex]\(-5\)[/tex].
- This gives the interval [tex]\((- \infty, -5]\)[/tex].
2. For [tex]\( -2 < x < 4 \)[/tex]:
- The function [tex]\( f(x) = -\frac{3}{2} x + 3 \)[/tex] is defined for [tex]\( x \)[/tex] strictly between [tex]\(-2\)[/tex] and [tex]\(4\)[/tex].
- This gives the interval [tex]\((-2, 4)\)[/tex].
3. For [tex]\( x \geq 5 \)[/tex]:
- The function [tex]\( f(x) = -\frac{1}{2} x + 7 \)[/tex] is defined for all values of [tex]\( x \)[/tex] that are greater than or equal to [tex]\(5\)[/tex].
- This gives the interval [tex]\([5, \infty)\)[/tex].
Next, we combine these intervals to get the domain of the entire piecewise function [tex]\( f(x) \)[/tex]. The combined intervals are:
- [tex]\((- \infty, -5]\)[/tex]
- [tex]\((-2, 4)\)[/tex]
- [tex]\([5, \infty)\)[/tex]
However, the function does not cover the intervals [tex]\((- \infty, -5]\)[/tex], [tex]\([-5, -2]\)[/tex], and [tex]\([4, 5]\)[/tex]. Properly written, we have:
[tex]\[
\text{Domain of } f(x) = (- \infty, -5] \cup (-2, 4) \cup [5, \infty)
\][/tex]
This indicates that the function [tex]\( f(x) \)[/tex] is defined for:
- All [tex]\( x \)[/tex] less than or equal to [tex]\(-5\)[/tex]
- All [tex]\( x \)[/tex] strictly between [tex]\(-2\)[/tex] and [tex]\(4\)[/tex]
- All [tex]\( x \)[/tex] greater than or equal to [tex]\(5\)[/tex]
Thus, the domain of [tex]\( f(x) \)[/tex] is [tex]\((- \infty, -5] \cup (-2, 4) \cup [5, \infty)\)[/tex].
