We have to evaluate the piecewise defined function
[tex]g(x)=\mleft\{\begin{aligned}-2\text{ }if\text{ x<-2} \\ (x-1)^2-2\text{ if -2}\leq x\leq2 \\ -\frac{1}{2}x+2\text{ if x>2}\end{aligned}\mright.[/tex]
to evaluate this type of function we have to be careful to choose the right relation for the number we are evaluating.
Let's do the first one
[tex]g(-2)[/tex]
Since the -2 lies in the second interval of the function we have to choose the second relation in the function, then
[tex]\begin{gathered} g(-2)=(-2-1)^2-2 \\ =(-3)^2-2 \\ =9-2 \\ =7 \end{gathered}[/tex]
therefore g(-2)=7.
To evaluate g(0) we notice that x=0 also lies in the second interval of the piecewise function. Then
[tex]\begin{gathered} g(0)=(0-1)^2-2 \\ =(-1)^2-2 \\ =1-2 \\ =-1 \end{gathered}[/tex]
therefore g(0)=-1.
Finally, to evaluate g(5) we notice that x=5 lies in the third interval of the function, then
[tex]\begin{gathered} g(5)=-\frac{1}{2}(5)+2 \\ =-\frac{5}{2}+2 \\ =-\frac{1}{2} \end{gathered}[/tex]
Therefore g(5)=-1/2.
