Answer: The required value of the given expression is (h +1).
Step-by-step explanation: We are given a function f(x) as follows :
[tex]f(x)=x^2-3x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to evaluate the following :
[tex]E=\dfrac{f(2+h)-f(2)}{h}.[/tex]
Substituting x = 2 + h in equation (i), we get
[tex]f(2+h)\\\\=(2+h)^2-3(2+h)\\\\=(4+4h+h^2)-6-3h\\\\=4+4h+h^2-6-3h\\\\=h^2+h-2[/tex]
and substituting x = 2 in equation (i), we get
[tex]f(2)=2^2-3\times2=4-6=-2.[/tex]
Therefore, the value of expression E can be evaluated as follows :
[tex]E\\\\\\=\dfrac{f(2+h)-f(2)}{h}\\\\\\=\dfrac{(h^2+h-2)-(-2)}{h}\\\\\\=\dfrac{h^2+h-2+2}{h}\\\\\\=\dfrac{h^2+h}{h}\\\\\\=\dfrac{h(h+1)}{h}\\\\=h+1.[/tex]
Thus, the required value of the given expression is (h +1).
