The graph shows a piecewise function, meaning that different functions are shows at different intervals (ranges of x-values).
1) One of the graphs is part of a parabola, meaning the equation is quadratic (has [tex] x^{2} [/tex]). That equation must be [tex] x^{2} + 4[/tex].
2) The other graph is a straight line, meaning it must be a linear equation. That equation must be x + 4.
Looking at the entire graph, you can see that the parabola starts from the left and ends with an open circle at x=2. An open circle means that the graph doesn't have a value at that point, x=2. The linear line starts with a filled point at x=2 and continues to the right. That means we're looking for the choice where:
1) [tex]x^{2} + 4 \ \textless \ 2[/tex]
Since x=2 cannot be a point in the graph, and less than 2 means that the graph includes everything to the left of 2, but not including 2
2) [tex]x + 4 \geq 2[/tex]
Since x=2 is a point in the graph, and greater than or equal to 2 means that the graph includes 2 and everything to the right of it
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Answer: C) [tex]y = \left \{ {{x^{2} + 4 \textless \ 2} \atop {x + 4 \: \geq \: 2}} \right. [/tex]
