Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Domain and Range of the Real Valued Function: (Ii) F ( X ) = a X − B C X − D - Mathematics

Find the domain and range of the real valued function:

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

#### Solution

Given:

$f\left( x \right) = \frac{ax - b}{cx - d}$
Domain of f : Clearly,  (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 ( cya) = 0, i.e. cy = a.
$\Rightarrow y = \frac{a}{c}$ Hence, range ( f ) = $R - \left\{ \frac{a}{c} \right\}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 3 Functions
Exercise 3.3 | Q 3.02 | Page 18