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

Find the domain and range of the real valued function:

(i) $f\left( x \right) = \frac{ax + b}{bx - a}$

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#### Solution

(i)
Given:

$f\left( x \right) = \frac{ax + b}{bx - a}$

Domain of f : Clearly,  (x) is a rational function of x as

$\frac{ax + b}{bx - a}$ is a rational expression.

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

$\Rightarrow x = \frac{a}{b}$
Hence, domain ( f ) =$R - \left\{ \frac{a}{b} \right\}$
Range of f :
Let f (x) = y ⇒ (ax + b) = y (bx -a)
⇒ (ax + b) = (bxy -ay)
⇒ b + ay = bxy -ax
⇒ b + ay = x(by - a)

$\Rightarrow x = \frac{b + ay}{by - a}$
Clearly, f (x) assumes real values for all x except for all those values of x for which ( by - a) = 0, i.e. by = a.
$\Rightarrow y = \frac{a}{b}$
Hence, range ( f ) =$R - \left\{ \frac{a}{b} \right\}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 3 Functions
Exercise 3.3 | Q 3.01 | Page 18
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