The average rate of funtion is defined as :
[tex]\frac{f(b)-f(a)}{b-a}\to\mleft\lbrack a,b\mright\rbrack[/tex]
From the problem we have :
[tex]g(x)=4x^2-7\to\mleft\lbrack1,b\mright\rbrack[/tex]
We evaluate at each of the extremes of the interval
[tex]\begin{gathered} g(1)=4\cdot(1)^2-7 \\ g(1)=4-7 \\ g(1)=-3 \end{gathered}[/tex][tex]g(b)=4b^2-7[/tex]
We replace in the equation the average rate of change
[tex]\begin{gathered} \frac{(4b^2-7)-(-3)}{b-1}=\frac{4b^2-4}{b-1}=\frac{4(b^2-1)}{b-1} \\ \end{gathered}[/tex]
factoring
[tex]b^2-1=(b-1)(b+1)[/tex]
We simplify the average rate of change
[tex]\frac{(4b^2-7)-(-3)}{b-1}=\frac{4(b^{}-1)(b+1)}{b-1}=4(b+1)[/tex]
Ethnoces the answer is 4(b+1)
