Answer:
(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
(B) f(x + h) - f(x) = 8xh + 4h² - 6h
(C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
Step-by-step explanation:
* Lets explain how to solve the problem
- The function f(x) = 4x² - 6x + 6
- To find f(x + h) substitute x in the function by (x + h)
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) = 4(x + h)² - 6(x + h) + 6
- Lets simplify 4(x + h)²
∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²
∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²
- Lets simplify 6(x + h)
∵ 6(x + h) = 6(x) + 6(h)
∴ 6(x + h) = 6x + 6h
∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6
- Remember (-)(+) = (-)
∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
- Lets find f(x + h) - f(x)
∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)
- Remember (-)(-) = (+)
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6
- Simplify by adding the like terms
∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)
∴ f(x + h) - f(x) = 8xh + 4h² - 6h
* (B) f(x + h) - f(x) = 8xh + 4h² - 6h
- Lets find [tex]\frac{f(x+h)-f(x)}{h}[/tex]
∵ f(x + h) - f(x) = 8xh + 4h² - 6h
∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh + 4h^{2}-6h}{h}[/tex]
- Simplify by separate the three terms
∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh}{h}+\frac{4h^{2} }{h}-\frac{6h}{h}[/tex]
∴ [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
* (C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
