Question

Find the domain and range of the real valued function:

(ii) \[f\left( x \right) = \frac{ax - b}{cx - d}\]

 

 

Answer 1

Given:

\[f\left( x \right) = \frac{ax - b}{cx - d}\] 

Domain of f : Clearly,  f (x) is a rational function of x as \[\frac{ax - b}{cx - d}\] is a rational expression.

Clearly, f (x) assumes real values for all x except for all those values of x for which ( cx - d) = 0, i.e. cx = d.

\[\Rightarrow x = \frac{d}{c}\] Hence, domain ( f ) = \[R - \left\{ \frac{d}{c} \right\}\] Range of f :
Let f (x) = y ⇒ (ax -b) = y( cx -d)
⇒ (ax - b) = (cxy - dy)

⇒ dy - b = cxy - ax 

⇒ dy  - b = x(cy - a)

 \[\Rightarrow x = \frac{dy - b}{cy - a}\]

Clearly, f (x) assumes real values for all x except for all those values of x for which ( cy - a) = 0, i.e. cy = a.

\[\Rightarrow y = \frac{a}{c}\] Hence, range ( f ) = \[R - \left\{ \frac{a}{c} \right\}\]