Question

If \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be functions such that \(f(x)=\cos x\) and \(g(x)=3x^3\), then find \(f\circ g\).

Hint:

\((f\circ g)(x)=f(g(x))\): replace the input of \(\cos\) with \(3x^3\).

Answer 1

Step 1: Symbol-substitution check:
Write \(f(t)=\cos t\) as a rule in a dummy variable \(t\). Then \((f\circ g)(x)=f(t)\big|_{t=g(x)}\).

Step 2: Substituting t = g(x):
With \(t=3x^3\), \(f(t)=\cos t\) becomes \(\cos(3x^3)\).

Final Answer:
Confirms \((f\circ g)(x)=\boxed{\cos(3x^3)}\).