Question

Find and simplify each of the following for f(x) = 4x2 - 5x + 7.(A) f(x + h)(B) f(x +h)-f(x)f(x+h) – f(x)(C)h

Answer 1

`f(x)=4x^2\text{ - 5x + 7}`

(A) f(x+h)

`\begin{gathered} f(x+h)=4(x+h)^2\text{ - 5(x + h) + 7} \\ =\text{ 4(x + h)(x + h) - 5x - 5h + 7} \\ =4(x^2+2xh+h^2)\text{ - 5x - 5h + 7} \\ =4x^2+8xh+4h^2\text{ - 5x - 5h + 7} \end{gathered}`

(B) f(x + h) - f(x)

`\begin{gathered} =4x^2+8xh+4h^2\text{ - 5x - 5h + 7 - }(4x^2\text{ - 5x + 7)} \\ =4x^2+8xh+4h^2-5x-5h+7-4x^2\text{ + 5x - 7} \\ =4x^2-4x^2-5x+5x+7-7+8xh+4h^2\text{ - 5h} \\ \text{ = 0 + 0 + 0 + 8xh + 4h}^2\text{ - 5h} \\ =8xh+4h^2\text{ - 5h} \end{gathered}`

(C)

`\begin{gathered} =\text{ }\frac{f(x+h)\text{ - f(h)}}{h} \\ =\text{ }\frac{8xh+4h^2\text{ - 5h}}{h} \\ =\text{ }(8xh)/(h)\text{ + }(4h^2)/(h)\text{ - }(5h)/(h) \\ =\text{ 8x + 4h - 5} \end{gathered}`