Question:
If functions \(f:R\to R\) and \(g:R\to R\) are defined respectively as \(f(x)=\cos x\) and \(g(x)=3x^2\), then find \(gof\) and \(fog\).
If functions \(f:R\to R\) and \(g:R\to R\) are defined respectively as \(f(x)=\cos x\) and \(g(x)=3x^2\), then find \(gof\) and \(fog\).
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Substitute f into g for gof, and g into f for fog, in that order.
Updated On: Sep 23, 2026
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Solution and Explanation
Step 1: Compute \(gof(x)=g(f(x))\):
\(f(x)=\cos x\), so \(g(f(x))=g(\cos x)=3(\cos x)^2=3\cos^2x\).
Step 2: Compute \(fog(x)=f(g(x))\):
\(g(x)=3x^2\), so \(f(g(x))=f(3x^2)=\cos(3x^2)\).
Final Answer:
\[ \boxed{gof(x)=3\cos^2x,\qquad fog(x)=\cos(3x^2)} \]
\(f(x)=\cos x\), so \(g(f(x))=g(\cos x)=3(\cos x)^2=3\cos^2x\).
Step 2: Compute \(fog(x)=f(g(x))\):
\(g(x)=3x^2\), so \(f(g(x))=f(3x^2)=\cos(3x^2)\).
Final Answer:
\[ \boxed{gof(x)=3\cos^2x,\qquad fog(x)=\cos(3x^2)} \]
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