Question

Find range of:

f(x) = {x+4 if -2 <= x < 1
8 if x = 1
-x+3 if x> 1

Answer 1

`f(x)=\left\{\begin{array}{ccc}x+4&if&-2\leq x < 1\\8&if&x=1\\-x+3&if&x > 1\end{array}\right\\\\(1)\ y=x+4\ for\ x\in[-2,\ 1)\\\\\text{It's a linear function.}\\\text{It's enough to calculate the function values at the ends of the interval.}\\\\for\ x=-2\to y=-2+4=2\\for\ x=1\to y=1+4=5\\\\y\in[2,\ 5)\\\\(2)\ y=8\ for\ x=1\\\\\text{It's only one value of y.}\\\\y\in\{8\}\\\\(2)\ y=-x+3\ for\ x\in(1,\ \infty)\\\\\text{It's a linear function.}\\\text{It's decreasing function because the slope}\ m = -1 < 0.`

`\text{Therefore for}\ x\to\infty,\ y\to-\infty.\\\\for\ x=1\to y=-1+3=2\\\\y\in(-\infty,\ 2).\\\\From\ (1),\ (2)\ and\ (3)\ we\ have\ the\ answer:\\\\Range:y\in(-\infty,\ 5)\ \cup\ \{8\}`