The given function is:
[tex]f(x)\text{ = }x^2\text{ + 3}[/tex][tex]x_1=3,x_2=\text{ 4}[/tex][tex]\begin{gathered} f(x_1)=f(3)=3^2+3\text{ = 9 + 3 } \\ f(x_1)\text{ = f(3) = 1}2 \end{gathered}[/tex][tex]\begin{gathered} f(x_2)=f(4)=4^2+3\text{ = 16 + 3 } \\ f(x_2)\text{ = f(4) = 19} \end{gathered}[/tex]The average rate of change is given by the formula for a slope:
[tex]\begin{gathered} \frac{df(x)}{dx}=\text{ }\frac{f(x_2)-f(x_1)}{x_2-x_1} \\ \frac{df(x)}{dx}=\text{ }\frac{f(4)-f(3)}{4-3} \\ \frac{df(x)}{dx}=\text{ }\frac{19-12}{4-3} \\ \frac{df(x)}{dx}=\text{ }\frac{7}{1} \\ \frac{df(x)}{dx}=\text{ 7} \end{gathered}[/tex]The average rate of change = 7 units