Gujarat BoardEnglish MediumSTD 11 ScienceMATHSFunctions2 Marks
Question
Find the domain and range of the following real valued functions: $\text{f(x)}=\frac{\text{ax}-\text{b}}{\text{cx}-\text{d}}$
✓
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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