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

Answer 1

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