Answer:
[tex]\displaystyle f(x)=\left \{ \begin{array}{ll}{{-(x+3)^2+3&\text{for $x<-2$}} \\ {\dfrac{1}{2}|x|-1}&\text{for $x\ge -2$}}\end{array} \right.[/tex]
Step-by-step explanation:
The left section of the function is a parabola with vertex (-3, 3) and a vertical scale factor of -1. You can tell that -1 is the scale factor because the parabola opens downward and 1 horizontal unit from the vertex, the function is 1 unit vertically different from the vertex. (The vertical difference is the scale factor.)
The right section is an absolute value function with a vertex at (0, -1) and a scale factor of 1/2. You can tell the scale factor is 1/2 because the rise/run of the lines is 1/2.
The left section is defined for x < -2; the right section is defined for x ≥ -2.
The vertex locations show the function translation, which is applied in the usual way:
f(x) translated to (h, k) is f(x -h) +k.
[tex]\displaystyle f(x)=\left \{ \begin{array}{ll}{{-(x+3)^2+3&\text{for $x<-2$}} \\ {\dfrac{1}{2}|x|-1}&\text{for $x\ge -2$}}\end{array} \right.[/tex]
