1. Let \[f(x) = \left\{ \begin{array}{cl} 2x + 1 & \text{if } x \le 3, \\ 8 - 4x & \text{if } x > 3. \end{array} \right.\]Find the sum of all values of $x$ such that $f(x) = 0.$
2. Find the domain of $\frac{x^2 + 10x + 21}{x^2 + 4x - 21}$. (Express your answer using interval notation.)
\(f(x) = \left\{ \begin{array}{cl} 2x + 1 & \text{if } x \le 3, \\ 8 - 4x & \text{if } x > 3. \end{array} \right.\ \)
Find the sum of all values of x such that f(x) = 0
0 = 2x + 1 subtract 1 from each side
-1 = 2x divide both sides by 2
-1/2 = x
And
0 = 8 - 4x add 4x to both sides
4x = 8 divede both sides by 4
x = 2 [ not in the domain of the second function ]
So the sum of all values such that f(x) = 0 is just -1/2
2. Find the domain of {x^2 + 10x + 21}/{x^2 + 4x - 21}. (Express your answer using interval notation.)
The domain will only be restricted by all the values of x that make the denominator = 0
So we have
x^2 + 4x - 21 = 0 factor
(x + 7) ( x - 3) = 0
Set both factors to 0 and solve for x and we havethat x = -7 and x = 3
So....the domain is (-infinity, -7) U ( -7, 3) U (3, infinity)