Question

5. Suppose f(x) = x² + x - 1, simplify the following:(a) f(x + h) =(b) f(x+h) - f(x) =(c) Compute the difference quotient(f(x+h)-f(x))/hSimplify until the h in the denominator is canceled

Answer 1

(a) Here we just need to switch every x by "x + h", so we have:

`\begin{gathered} f(x)=x^2+x-1 \\ f(x+h)=(x+h)^2+(x+h)-1 \\ f(x+h)=x^2+2xh+h^2+x+h-1 \end{gathered}`

(b) Subtracting the previous expression from f(x), we have:

`\begin{gathered} f(x+h)-f(x)=x^2+2xh+h^2+x+h-1-(x^2+x-1) \\ f(x+h)-f(x)=2xh+h^2+h \end{gathered}`

(c) Now, dividing the previous result by h, we have:

`(f(x+h)-f(x))/(h)=(2xh+h^2+h)/(h)=2x+h+1`