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

Therefore, the difference quotient is;

`(f(x+h)-f(x))/(h)=-2x-h`