Let
\(f(n) = \left\{ \begin{array}{cl} n^2-2 & \text{ if }n<0, \\ 2n-20 & \text{ if }n \geq 0. \end{array} \right.\)
What is the positive difference between the two values of a that satisfy the equation \(f(-2)+f(2)+f(a)=0\)
+23245 f(-2) = (-2)2 - 2 = 4 - 2 = 2
f(2) = 2(2) - 20 = 4 - 20 = -16
There are two possible conditions on a: either a < 0 or a >= 0.
If a < 0: f(a) = a2 - 2 ---> f(-2) + f(2) + f(a) = 0
---> 2 + -16 + (a2 - 2) = 0
---> a2 - 16 = 0
---> (a - 4)(a + 4) = 0
So, either a = 4 or a = -4 but this condition says that a < 0, so the value of a is -4
If a >= 0: f(a) = a2 - 2 ---> f(-2) + f(2) + f(a) = 0
---> 2 + -16 + (2a - 20) = 0
---> 2a - 34 = 0
---> 2a = 34
---> a = 17
The positive difference is: 17 - -4 = 21
