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 \(fog\).

Hint:

Substitute f into g for gof, and g into f for fog, in that order.

Answer 1

Step 1: Recall composition order — gof means apply f first:
$(g\circ f)(x)=g(f(x))$: feed $f(x)=\cos x$ into $g(t)=3t^2$, giving $3(\cos x)^2=3\cos^2x$.

Step 2: fog means apply g first:
$(f\circ g)(x)=f(g(x))$: feed $g(x)=3x^2$ into $f(t)=\cos t$, giving $\cos(3x^2)$.

Final Answer:
\[ \boxed{gof(x)=3\cos^2x,\ \ fog(x)=\cos(3x^2)} \]