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Question
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
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Solution
Here f(x) = |x| + x which can be redefined as
f(x) = `{(2x, "if" x ≥ 0),(0, "if" x < 0):}`
Similarly, the function g defined by g(x) = |x| – x may be redefined as
g(x) = `{(0, "if" x ≥ 0),(-2x, "if" x < 0):}`
Therefore, g o f gets defined as:
For x ≥ 0, (g o f) (x) = g (f(x) = g (2x) = 0
and for x < 0, (g o f) (x) = g (f(x) = g (0) = 0.
Consequently, we have (g o f) (x) = 0, ∀ x ∈ R.
Similarly, f o g gets defined as:
For x ≥ 0, (f o g) (x) = f (g(x) = f(0) = 0,
and for x < 0, (f o g) (x) = f (g(x)) = f(–2x) = – 4x.
i.e. (f o g) (x) = `{(0, x > 0),(-4x, x < 0):}`
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