Question

f(x+h)-f(x)Find the difference quotienthwhere hx0, for the function below.f(x) = -x + 2Simplify your answer as much as possible.[(x + n) - (*)D음h

Answer 1

Given:

$f\mleft(x\mright)=-x^2+2$

Hence, f(x+h) will be;

$\begin{gathered} f(x+h)=-(x+h)^2+2 \\ =-(x^2+2xh+h^2)+2 \\ f(x+h)=-x^2-2xh-h^2+2 \\ \end{gathered}$

The difference quotient of the function will be;

$\begin{gathered} \frac{f(x+h)-f(x)}{h} \\ \text{SUbstituting f(x) and f(x+h) in the above formula;} \\ =\frac{-x^2-2xh-h^2+2-(-x^2+2)}{h} \\ =\frac{-x^2-2xh-h^2+2+x^2-2}{h} \\ =\frac{-x^2+x^2-2xh-h^2+2-2}{h} \\ =\frac{-2xh-h^2}{h} \\ \text{Factorizing the common factor out;} \\ =\frac{h(-2x-h)}{h} \\ \frac{f(x+h)-f(x)}{h}=-2x-h \end{gathered}$

Therefore, the difference quotient is;

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