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Question
If f(x) = loge (1 − x) and g(x) = [x], then determine function:
(ii) fg
Solution
Given:
f(x) = loge (1 − x) and g(x) = [x]
Clearly, f(x) = loge (1 − x) is defined for all ( 1 -x) > 0.
⇒ 1 > x
⇒ x < 1
⇒ x ∈ ( -∞, 1)
Thus, domain (f ) = ( - ∞, 1)
Again,
g(x) = [x] is defined for all x ∈ R.
Thus, domain (g) = R
∴ Domain (f) ∩ Domain (g) = ( - ∞, 1) ∩ R = ( -∞, 1)
Hence,
(ii) (fg) : ( - ∞, 1) → R is given by (fg) (x) = f(x).g(x) = loge (1 − x)[ x ] = [ x ]loge (1 − x).
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