problems 31—32 you are given the definition of function f. find the difference quotient (f(x + h)-f(x))/h…

problems 31—32 you are given the definition of function f. find the difference quotient (f(x + h)-f(x))/h, simplify it, and write the result in the box. example: f(x)=5x - 2 (f(x + h)-f(x))/h = 5 example: f(x)=7x² + 11x + 3 (f(x + h)-f(x))/h = 14x + 7h + 11 31. f(x)=4x + 6 (f(x + h)-f(x))/h = 32. f(x)=x² - 9x + 2 (f(x + h)-f(x))/h =

problems 31—32 you are given the definition of function f. find the difference quotient (f(x + h)-f(x))/h, simplify it, and write the result in the box. example: f(x)=5x - 2 (f(x + h)-f(x))/h = 5 example: f(x)=7x² + 11x + 3 (f(x + h)-f(x))/h = 14x + 7h + 11 31. f(x)=4x + 6 (f(x + h)-f(x))/h = 32. f(x)=x² - 9x + 2 (f(x + h)-f(x))/h =

Answer

Explanation:

Step1: Find $f(x + h)$ for $f(x)=4x + 6$

$f(x + h)=4(x + h)+6=4x+4h + 6$

Step2: Calculate $f(x + h)-f(x)$

$f(x + h)-f(x)=(4x + 4h+6)-(4x + 6)=4h$

Step3: Find the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{4h}{h}=4$

Step4: For $f(x)=x^{2}-9x + 2$, find $f(x + h)$

$f(x + h)=(x + h)^{2}-9(x + h)+2=x^{2}+2xh+h^{2}-9x-9h + 2$

Step5: Calculate $f(x + h)-f(x)$

$f(x + h)-f(x)=(x^{2}+2xh+h^{2}-9x-9h + 2)-(x^{2}-9x + 2)=2xh+h^{2}-9h$

Step6: Find the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}-9h}{h}=2x+h - 9$

Answer:

  1. 4
  2. $2x+h - 9$