Question

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

(iii) \[\frac{f}{g}\]

 

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,

(iii) Given:
  g(x) = [ x ]

  If  [ x ]  = 0,
 x ∈ (0, 1)
Thus,

\[\text{ domain } \left( \frac{f}{g} \right) = \text{ domain } \left( f \right) \cap \text{ domain } \left( g \right) - \left\{ x: g\left( x \right) = 0 \right\}\]

\[\frac{f}{g}: \left( - \infty , 0 \right) \to \text{ R is defined by } \left( \frac{f}{g} \right)\left( x \right) = \frac{f\left( x \right)}{g\left( x \right)} = \frac{\log_e \left( 1 - x \right)}{\left[ x \right]} . \]