Answer to Question #289206 in Algebra for bookaddict

Question

f(x)=x^2-2x-3 for x>=0, -a<=x<=a. if the range of the function f is -4<=f(x)<=5, find the value of a.

Answer 1

f(x) = x²-2x-3

-4 ≤ f(x) ≤ 5

x²-2x-3 ≥ -4, (1)

x²-2x-3 ≤ 5; (2)

(1) x²-2x-3 ≥ -4

x²-2x +1 ≥ 0

(x-1)² ≥ 0

x $\in$ (-∞, +∞)

(2) x²-2x-3 ≤ 5

x²-2x-8 ≤ 0

(x-4)(x+2) ≤ 0

x $\in$ [-2, 4]

(-∞, +∞) ∩ [-2, 4] = [-2, 4]

If -a≤x≤a, then a = 2, so x ∈ [-2;2]

ANSWER: a = 2