Find the domain and range of the following real valued functions:…

Question

Find the domain and range of the following real valued functions:
$\text{f(x)}=\frac{\text{ax}-\text{b}}{\text{cx}-\text{d}}$

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Answer

Given,
$\text{f(x)}=\frac{\text{ax}-\text{b}}{\text{cx}-\text{d}}$
Domain of f: Clearly, f(x) is a rational function of x as $\frac{\text{ax}-\text{b}}{\text{cx}-\text{d}}$ 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
$\Rightarrow\ \text{x}=\frac{\text{d}}{\text{c}}$
Hence, domain $(\text{f})=\text{R}-\Big\{\frac{\text{d}}{\text{c}}\Big\}$
Range of f,
Let f(x) = y
$\Rightarrow\ \frac{\text{ax}-\text{b}}{\text{cx}-\text{d}}=\text{y}$
$\Rightarrow\ (\text{ax}-\text{b})=\text{y}(\text{cx}-\text{d})$
$\Rightarrow\ (\text{ax}-\text{b})=(\text{cxy}-\text{dy})$
$\Rightarrow\ \text{dy}-\text{b}=\text{cxy}-\text{ax}$
$\Rightarrow\ \text{dy}-\text{b}=\text{x}(\text{cy}-\text{a})$
$\Rightarrow\ \text{x}=\frac{\text{dy}-\text{b}}{\text{cy}-\text{a}}$
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.
$\Rightarrow\ \text{y}=\frac{\text{a}}{\text{c}}$
Hence, range $(\text{f})=\text{R}-\Big\{\frac{\text{a}}{\text{c}}\Big\}$

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