A piecewise function is a function built from pieces of different functions over different intervals; in this case we have a piecewise function defined by two pieces.
The first piece tells us that if x is any value less than or equal to 3 the we need to use the expression 2x for the function.
The second piece tells us that if x is any value greater than 3 we have to use the expression:
[tex]\frac{1}{3}x^2-2x+9[/tex]
Now, to graph this function we need to make a table of values like any other function taking into acount on which interval is x in orther to use the proper expression for the output of the function:
In this table we use what we stated above; for example since 0 is less than 3 we have to use the expression 2x, then we have:
[tex]f(0)=2(0)=0[/tex]
and so on for the values one, two and three.
For the other values we notice that they are greater than 3 then we hace to use the second expression, for example:
[tex]\begin{gathered} f(4)=\frac{1}{3}(4)^2-2(4)+9 \\ f(4)=\frac{16}{3}-8+9 \\ f(4)=5.33\bar{3}-8+9 \\ f(4)=6.33\bar{3} \end{gathered}[/tex]
and so on for any value greater than 3.
Now we plot this points and join them to get the graph:
