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Question
If f, g : R → R are defined by f(x) = |x| + x and g(x) = |x| – x find g o f and f o g
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Solution
f(x) = |x| + x = `{{:(x + x = 2x, "if" x ≥ 0),(- x + x = 0, "if" x < 0):}`
g(x) = |x| – x = `{{:(x - x = 0, "if" x ≥ 0),(- x - x = - 2x, "if" x < 0):}`
f o g(x) = f(g(x)) = `{{:(f(0), "if" x ≥ 0),(f(- 2x), "if" x < 0):}`
f o g(x) = `{{:(2 xx 0 = 0, "if" x ≥ 0),(- x + 2x = 0, "if" x < 0):}`
∴ f o g(x) = 0 for all x ∈ R
g o f(x) = g(f(x)) = `{{:(g(2x), "if" x ≥ 0),(g(0), "if" x < 0):}`
g o f(x) = `{{:(0, "if" x ≥ 0),(0, "if" x < 0):}`
⇒ g of(x) = 0 for all x ∈ R
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