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Given \(f(x)=\frac{x-2}{x+3}\) and \(g(x)=\frac{x+3}{x-2}\). Find the domain of \((f g)(x)\).दिया है \(f(x)=\frac{x-2}{x+3}\) और \(g(x)=\frac{x+3}{x-2}\)। \((f g)(x)\) का डोमेन क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: All real numbers except \(x=-3\) and \(x=2\)सभी वास्तविक संख्याएँ सिवाय \(x=-3\) और \(x=2\) के
\((fg)(x)=f(x)g(x)\) is defined only when both fractions are defined.
For \(f(x)=\frac{x-2}{x+3}\), denominator \(x+3\neq 0\Rightarrow x\neq -3\).
For \(g(x)=\frac{x+3}{x-2}\), denominator \(x-2\neq 0\Rightarrow x\neq 2\).
So the domain is \(\mathbb{R}\setminus\{-3,2\}\).\((fg)(x)=f(x)g(x)\) तभी परिभाषित है जब दोनों भिन्नों का मान परिभाषित हो।
\(f(x)=\frac{x-2}{x+3}\) के लिए हर (denominator) \(x+3\neq 0\Rightarrow x\neq -3\)।
\(g(x)=\frac{x+3}{x-2}\) के लिए हर \(x-2\neq 0\Rightarrow x\neq 2\)।
इसलिए डोमेन \(\mathbb{R}\setminus\{-3,2\}\) है।
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