Question

for the function defined by g(x)=4x^2+3calculate and simplify the difference quotient g(x+h)-g(x)_________ h

Answer 1

g (x ) = 4x^2 + 3

g ( x + h ) = 4 ( x + h ) ^ 2 + 3

Now, we'll put all into the expression as follows:

$\begin{gathered} \frac{g\text{ ( x + h ) - g ( x )}}{h\text{ }} \\ =\text{ }\frac{4(x+h)^2-(4x^2\text{ + 3 )}}{h} \\ weneedtoexpand(x+h)^2 \\ (\text{ x + h )( x + h )} \\ x\text{ ( x + h ) + h ( x + h )} \\ x^2+xh+xh+h^2 \\ x^2+2xh+h^2 \\ \text{Put the value back into the equation} \\ \frac{(4(x^2+2xh+h^2)+3)-(4x^2\text{ + 3 )}}{h} \\ =\text{ }\frac{4x^2+8xh+4h^2-4x^2\text{ - 3}}{h}\text{ ( expanding )} \\ =\frac{4x^2-4x^2+8xh+4h^2\text{ - 3}}{h}\text{ ( like terms collected )} \\ =\text{ }\frac{8xh+4h^2\text{ + 3}}{h}\text{ ( since }4x^2-4x^2\text{ = 0 )} \end{gathered}$

The answer can be further sinplified in one step as follows:

$\begin{gathered} \frac{8xh}{h}\text{ + }\frac{4h^2}{h}\text{ + }\frac{3}{h} \\ =\text{ 8x + 4h + }\frac{3}{h} \end{gathered}$

Above is the simplification of the function