Question

Does anyone understand how to solve this?

Answer 1

You solve it by filling in the function value and simplifying the result.

a.

`\displaystyle (f(x)-f(a))/(x-a)=(\left(3x^(2)+x+3\right)-\left(3a^(2)+a+3\right))/(x-a)\\\\=(3\left(x^(2)-a^(2)\right)+\left(x-a\right))/(x-a)\\\\=(3(x+a)(x-a)+(x-a))/(x-a)=3(x+a)+1`

b.

`\displaystyle (f(x+h)-f(x))/(h)=(\left(3(x+h)^(2)+(x+h)+3\right)-\left(3x^(2)+x+3\right))/(h)\\\\=(\left(3\left(x^(2)+2hx+h^(2)\right)+(x+h)+3\right)-\left(3x^(2)+x+3\right))/(h)\\\\=(3\left(2hx+h^(2)\right)+h)/(h)=6x+1+3h`