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Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsFunctions2 Marks
Question
Find the domain and range of the following real valued functions:
$\text{f(x)}=\frac{\text{ax}+\text{b}}{\text{bx}-\text{a}}$
✓

Answer

Given,
$\text{f(x)}=\frac{\text{ax}+\text{b}}{\text{bx}-\text{a}}$
Domain of f: Clearly, f(x) is a rational function of x as $\frac{\text{ax}+\text{b}}{\text{bx}-\text{a}}$ 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
$\Rightarrow\ \text{x}=\frac{\text{a}}{\text{b}}$
Hence, domain $(\text{f})=\text{R}-\Big\{\frac{\text{a}}{\text{b}}\Big\}$
Range of f,
Let f(x) = y
$\Rightarrow\ \frac{\text{ax}+\text{b}}{\text{bx}-\text{a}}=\text{y}$
$\Rightarrow\ (\text{ax}+\text{b})=\text{y}(\text{bx}-\text{a})$
$\Rightarrow\ (\text{ax}+\text{b})=(\text{bxy}-\text{ay})$
$\Rightarrow\ \text{b}+\text{ay}=\text{bxy}-\text{ax}$
$\Rightarrow\ \text{b}+\text{ay}=\text{x}(\text{by}-\text{a})$
$\Rightarrow\ \text{x}=\frac{\text{b}+\text{ay}}{\text{by}-\text{a}}$
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.
$\Rightarrow\ \text{y}=\frac{\text{a}}{\text{b}}$
Hence, range $(\text{f})=\text{R}-\Big\{\frac{\text{a}}{\text{b}}\Big\}$

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