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Let f(x) = x + 1 for x <= 1 and f(x) = 2x + 1 for 1 < x <= 2. Let g(x) = x² for -1 <= x < 2 and g(x) = x + 2 for 2 <= x <= 3. Find the composite function (f o g)(x) = f(g(x)) together with its domain.

1. f(g(x)) = x² + 1 for -1 <= x <= 1, and 2x² + 1 for 1 < x < sqrt(2)
2. f(g(x)) = x² + 1 for all -1 <= x < 2
3. f(g(x)) = 2x² + 1 for -1 <= x <= 1, and x² + 1 for 1 < x < sqrt(2)
4. f(g(x)) = x + 3 for 2 <= x <= 3 only

Correct answer: f(g(x)) = x² + 1 for -1 <= x <= 1, and 2x² + 1 for 1 < x < sqrt(2)

Solution

For a composite f(g(x)), the input to f is g(x). You must restrict x so that g(x) falls in the domain of f (here x in (-1...) maps via x²), then apply the matching piece of f.

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