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Let f(x)= { 2x+a, x\ge -1 { bx^2+3, x<-1 and g(x)= { x+4, 0\le x\le 4 { -3x-2, -2<x<0 If the domain of the composite function g(f(x)) is [-1,4], then which of the following is true?

1. a = 0, b > 5
2. a = 2, b > 7
3. a = 2, b > 10
4. a = 0, b \in R

Correct answer: a = 2, b > 10

Solution

The correct option is valid because for the composite function g(f(x)) to have a domain of [-1, 4], f(x) must output values within the range of g(x). Setting a = 2 ensures that f(-1) = 2(-1) + 2 = 0, which is within the domain of g, and b must be greater than 10 to ensure that f(x) for x < -1 does not exceed 4, maintaining the overall domain.

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