let f(x)=2x² - x - 4 and g(x)= - 3x + 7. find and simplify each of the following. 4. $\frac{f(3 +…

Explanation:

Step1: Find (f(3 + h))

Substitute (x = 3+h) into (f(x)=2x^{2}-x - 4). [ \begin{align*} f(3 + h)&=2(3 + h)^{2}-(3 + h)-4\ &=2(h^{2}+6h + 9)-3 - h-4\ &=2h^{2}+12h+18-3 - h-4\ &=2h^{2}+11h + 11 \end{align*} ]

Step2: Find (f(3))

Substitute (x = 3) into (f(x)=2x^{2}-x - 4). [ \begin{align*} f(3)&=2\times3^{2}-3 - 4\ &=2\times9-3 - 4\ &=18-3 - 4\ &=11 \end{align*} ]

Step3: Calculate (\frac{f(3 + h)-f(3)}{h})

[ \begin{align*} \frac{f(3 + h)-f(3)}{h}&=\frac{(2h^{2}+11h + 11)-11}{h}\ &=\frac{2h^{2}+11h}{h}\ &=\frac{h(2h + 11)}{h}\ &=2h+11, h\neq0 \end{align*} ]

Step4: Find (g(-1 + h))

Substitute (x=-1 + h) into (g(x)=-3x + 7). [ \begin{align*} g(-1 + h)&=-3(-1 + h)+7\ &=3-3h + 7\ &=10-3h \end{align*} ]

Step5: Find (g(-1))

Substitute (x=-1) into (g(x)=-3x + 7). [ \begin{align*} g(-1)&=-3\times(-1)+7\ &=3 + 7\ &=10 \end{align*} ]

Step6: Calculate (\frac{g(-1 + h)-g(-1)}{h})

[ \begin{align*} \frac{g(-1 + h)-g(-1)}{h}&=\frac{(10-3h)-10}{h}\ &=\frac{-3h}{h}\ &=-3, h\neq0 \end{align*} ]

Answer:

For (\frac{f(3 + h)-f(3)}{h}), the answer is (2h + 11) ((h\neq0)). For (\frac{g(-1 + h)-g(-1)}{h}), the answer is (-3) ((h\neq0)).