for the function $f(x)=3x - 6$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video…

for the function $f(x)=3x - 6$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video written example message instructor\nsubmit question\nfor the function $f(x)=2x^{2}-2x$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video written example message instructor\nsubmit question\nfor the function $f(x)=4x^{2}+5x$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video written example message instructor\nsubmit question

for the function $f(x)=3x - 6$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video written example message instructor\nsubmit question\nfor the function $f(x)=2x^{2}-2x$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video written example message instructor\nsubmit question\nfor the function $f(x)=4x^{2}+5x$, evaluate and simplify.\n$\frac{f(x + h)-f(x)}{h}$\nquestion help: video written example message instructor\nsubmit question

Answer

Answer:

  1. $3$
  2. $4x + 2h - 2$
  3. $8x + 4h + 5$

Explanation:

Question 16

Step1: Find (f(x + h))

Given (f(x)=3x - 6), then (f(x + h)=3(x + h)-6=3x+3h - 6)

Step2: Substitute into (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(3x + 3h-6)-(3x - 6)}{h}\ &=\frac{3x+3h - 6-3x + 6}{h}\ &=\frac{3h}{h}\ &=3 \end{align*} ]

Question 17

Step1: Find (f(x + h))

Given (f(x)=2x^{2}-2x), then (f(x + h)=2(x + h)^{2}-2(x + h)) [ \begin{align*} f(x + h)&=2(x^{2}+2xh+h^{2})-2x-2h\ &=2x^{2}+4xh + 2h^{2}-2x-2h \end{align*} ]

Step2: Substitute into (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(2x^{2}+4xh + 2h^{2}-2x-2h)-(2x^{2}-2x)}{h}\ &=\frac{2x^{2}+4xh+2h^{2}-2x - 2h-2x^{2}+2x}{h}\ &=\frac{4xh+2h^{2}-2h}{h}\ &=\frac{h(4x + 2h-2)}{h}\ &=4x + 2h-2 \end{align*} ]

Question 18

Step1: Find (f(x + h))

Given (f(x)=4x^{2}+5x), then (f(x + h)=4(x + h)^{2}+5(x + h)) [ \begin{align*} f(x + h)&=4(x^{2}+2xh+h^{2})+5x+5h\ &=4x^{2}+8xh+4h^{2}+5x + 5h \end{align*} ]

Step2: Substitute into (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(4x^{2}+8xh+4h^{2}+5x + 5h)-(4x^{2}+5x)}{h}\ &=\frac{4x^{2}+8xh+4h^{2}+5x+5h - 4x^{2}-5x}{h}\ &=\frac{8xh+4h^{2}+5h}{h}\ &=\frac{h(8x + 4h+5)}{h}\ &=8x + 4h+5 \end{align*} ]