Answer: The evaluations are done below.
Step-by-step explanation: We are given the following function :
[tex]f(x)=3x^2-5x+7.[/tex]
We are to find the value of the following expressions :
[tex](A)~f(x+h)\\\\(B)~f(x+h)-f(x)\\\\(C)~\dfrac{f(x+h)-f(x)}{h}[/tex]
To find the above expressions, we must use the given value of f(x) as follows :
[tex](A)~\textup{We have}\\\\f(x+h)\\\\=3(x+h)^2-5(x+h)+7\\\\=3(x^2+2xh+h^2)-5x-5h+7\\\\=3x^2+6xh+3h^2-5x-5h+7.[/tex]
[tex](B)~\textup{We have}\\\\f(x+h)-f(x)\\\\=(3x^2+6xh+3h^2-5x-5h+7)-(3x^2-5x+7)\\\\=6xh+3h^2-5h.[/tex]
[tex](C)~\textup{We have}\\\\\dfrac{f(x+h)-f(x)}{h}\\\\\\=\dfrac{6xh+3h^2-5h}{h}\\\\\\=\dfrac{h(6x+3h-5)}{h}\\\\=6x+3h-5.[/tex]
Thus, all the expressions are evaluated.
