Moderate -0.3 This question tests recall of standard definitions (odd/even functions) and requires a straightforward algebraic proof using those definitions. The proof involves simple substitution: gf(-x) = g(f(-x)) = g(-f(x)) = g(f(x)), which is a direct application of the definitions with no novel insight required. While it's a proof question, it's more routine than the average A-level question.
6 Write down the conditions for \(\mathrm { f } ( x )\) to be an odd function and for \(\mathrm { g } ( x )\) to be an even function.
Hence prove that, if \(\mathrm { f } ( x )\) is odd and \(\mathrm { g } ( x )\) is even, then the composite function \(\mathrm { gf } ( x )\) is even.
6 Write down the conditions for $\mathrm { f } ( x )$ to be an odd function and for $\mathrm { g } ( x )$ to be an even function.\\
Hence prove that, if $\mathrm { f } ( x )$ is odd and $\mathrm { g } ( x )$ is even, then the composite function $\mathrm { gf } ( x )$ is even.
\hfill \mbox{\textit{OCR MEI C3 2010 Q6 [4]}}