Question

Given f (x) = x2 + 3x + 4, what is f of the quantity 2 plus h end quantity minus f of 2 all over h equal to?

Answer 1

The function is given as,

$f(x)=x^2+3x+4$

It is asked to determine the value of the expression,

$\frac{f(2+h)-f(2)}{h}$

Substitute x = 2+h in the function,

$\begin{gathered} f(2+h)=(2+h)^2+3(2+h)+4 \\ f(2+h)=2^2+h^2+2(2)(h)+3(2)+3(h)+4 \\ f(2+h)=4+h^2+4h+6+3h+4 \\ f(2+h)=h^2+7h+14 \end{gathered}$

Substitute x = 2 in the function,

$\begin{gathered} f(2)=(2)^2+3(2)+4 \\ f(2)=4+6+4 \\ f(2)=14 \end{gathered}$

Substitute these values in the required expression,

$\begin{gathered} \frac{f(2+h)-f(2)}{h} \\ =\frac{(h^2+7h+14)-14}{h} \\ =\frac{h^2+7h+0}{h} \\ =\frac{h(h+7)}{h} \\ =h+7 \\ =7+h \end{gathered}$

Thus, the value of the expression is equal to

$7+h$

Therefore, 1st option is the correct choice.