Question

1. $k(x)=\

$$\begin{cases}x + 4&\\text{if }x < -1\\\\5&\\text{if }-1 < x < 2\\\\-\\dfrac{1}{2}x + 1&\\text{if }x \\geq 2\\end{cases}$$

$
$d = \underline{\quad\quad\quad\quad\quad\quad\quad\quad}$
$r = \underline{\quad\quad\quad\quad\quad\quad\quad\quad}$

Explanation:

Step1: Find the domain

The domain of a piece - wise function is the union of the intervals for each piece.
For \(y=x + 4\), the interval is \(x\lt-1\).
For \(y = 5\), the interval is \(-1\lt x\lt2\).
For \(y=-\frac{1}{2}x + 1\), the interval is \(x\geq2\).
The domain \(D\) is all real numbers except \(x=-1\). So \(D=(-\infty,-1)\cup(-1,\infty)\).

Step2: Find the range

For \(y=x + 4\) when \(x\lt-1\), \(y=x + 4\lt-1 + 4=3\).
For \(y = 5\) when \(-1\lt x\lt2\), \(y = 5\).
For \(y=-\frac{1}{2}x+1\) when \(x\geq2\), \(y=-\frac{1}{2}x + 1\leq-\frac{1}{2}\times2+1=0\).
The range \(R=(-\infty,3)\cup\{5\}\).

Answer:

\(D = (-\infty,-1)\cup(-1,\infty)\)
\(R=(-\infty,3)\cup\{5\}\)