Answer to Question #247680 in Calculus for Anino

Question

1. find the domains of f(x)= √x+7 / x+4= g(x) = 3 / |x| + 2 
2. Draw the graph of f(x) =x³. then use it together with some transformation to sketch the graph of g(x) =(x - 3)³ - 4.

Answer 1

1. Let us find the domains of the functions  $f(x)=\frac{ \sqrt{x+7}}{ x+4}$ and $g(x) = \frac{3}{|x| + 2}.$ The domain of function $f$ consists of all real numbers $x$ such that $x+7\ge0$ and $x\ne-4.$ Therefore, this domain is equal to $[-7,-4)\cup(-4,+\infty).$ Since $|x|+2>0$ for each $x\in\R,$ we conclude that the domain of the function $g$ is the set $\R$ of all real numbers.

2. Let us draw the graph of $f(x) =x^3:$

Then use it together with the shift 3 units to the right and 4 units downward to sketch the graph of the function $g(x) =(x - 3)^3 - 4:$