We have g(x) that is defined by parts:
[tex]g(x)\begin{cases}x+3,x\le4 \\ -2x+7,x>4\end{cases}[/tex]
The first part has a y-intercept at y=3 and a slope of 1.
We can find its value in the limit: at x=4.
[tex]g(4)=x+3=4+3=7[/tex]
Then, the point is (4,7).
We can find the x-intercept as:
[tex]\begin{gathered} g(x)=0=x+3 \\ x=-3 \end{gathered}[/tex]
So another point of the line is (-3,0).
Drawing a line that passes through those two points will be the graph for x+3.
Then, for g(x) when x>4, the slope is -2 and the y-intercept is 7.
We evaluate the line at x=4, although this point does not belong to the line in this case (as x>4):
[tex]g(4)=-2(4)+7=-8+7=-1[/tex]
The point is then (4,-1).
We can calculate another point of the line in order to be able to graph it.
For example, for x=8:
[tex]g(8)=-2(8)+7=-16+7=-9[/tex]
The point is then (8,9).
We can now graph both parts of g(x):
