We have been given a piece wise function in which one graph is of parabola and other one is of a straight line.
When we shift the graph of [tex]y=x^2[/tex] above by 6 units then the equation of the graph is
[tex]y=x^2+6[/tex].
Since, the given graph of the parabola full fills this criteria. Hence, the equation of the parabola is given by
[tex]y=x^2+6[/tex]
The graph of the parabola is up to x=3. At point x=3 there is an open circle. Hence, we must include less than sign.
Now, the equation of the line is given by [tex]y=-x+6[/tex]
We have a solid sign , hence we must include greater than equal to sign.
Therefore, the equation for the piece wise function is given by
[tex]y=\left\{\begin{matrix} x^2=6, x<3\\ \\ -x+6, x\geq 3 \end{matrix}\right.[/tex]
C is the correct option.
