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Let \(f:\mathbb{R}\to\mathbb{R}\) be an even function and \(g:\mathbb{R}\to\mathbb{R}\) be an odd function. Which statement about \(h(x)=f(g(x))\) is always true?मान लें कि \(f:\mathbb{R}\to\mathbb{R}\) एक सम फलन और \(g:\mathbb{R}\to\mathbb{R}\) एक विषम फलन है। तब \(h(x)=f(g(x))\) के बारे में कौन-सा कथन सदैव सत्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(h\) is an even function.\(h\) सम फलन है।
Since \(g(-x)=-g(x)\) and \(f\) is even, \(h(-x)=f(-g(x))=f(g(x))=h(x)\). Hence \(h\) is even, not necessarily constant. Exam tip: apply the parity of the inner function first in a composition.क्योंकि \(g(-x)=-g(x)\) और \(f\) सम है, इसलिए \(h(-x)=f(-g(x))=f(g(x))=h(x)\)। अतः \(h\) सम है; यह अनिवार्यतः अचर नहीं होता। परीक्षा टिप: संयुक्त फलन में पहले भीतर वाले फलन की सममिति लगाएँ।
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