Question

If f(x) = log_e (1 − x) and g(x) = [x], then determine function:

(ii) fg

Answer 1

Given:
f(x) = log_e (1 − x) and g(x) = [x]
Clearly, f(x) = log_e (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) = log_e (1 − x)[ x ] = [ x ]log_e (1 − x).