Question

your own separate lined paper!! dq = \frac{f(x + h)-f(x)}{h}. be sure to simplify your 2. f(x)=x^{2}+5x + 1 4. f(x)=\frac{1}{x - 4} be sure to show all work leading to your answers and (x + 3) 7. f(x)=\sqrt{x^{2}-4} ne the domain of f(g(x)). show all work!! 9. f(x)=\frac{1}{x} and g(x)=\sqrt{x - 1} ) and f(x)\
eq g(x)\
eq x. 12. h(x)=\sec^{3}x 14. h(x)=\sqrt{\cos(5x)}

Explanation:

Step1: Find $f(x + h)$ for $f(x)=x^{2}+5x + 1$

$f(x + h)=(x + h)^{2}+5(x + h)+1=x^{2}+2xh+h^{2}+5x + 5h+1$

Step2: Substitute $f(x + h)$ and $f(x)$ into the difference - quotient formula

$DQ=\frac{(x^{2}+2xh+h^{2}+5x + 5h + 1)-(x^{2}+5x + 1)}{h}$

Step3: Simplify the numerator

$(x^{2}+2xh+h^{2}+5x + 5h + 1)-(x^{2}+5x + 1)=x^{2}+2xh+h^{2}+5x + 5h + 1 - x^{2}-5x - 1=2xh+h^{2}+5h$

Step4: Simplify the difference - quotient

$DQ=\frac{2xh+h^{2}+5h}{h}=\frac{h(2x + h+5)}{h}=2x+h + 5$

Answer:

$2x+h + 5$