Even/odd function verification

Questions asking students to prove algebraically whether a given function is even, odd, or neither, sometimes with accompanying sketches.

2 questions

OCR MEI C3 Q2
2
  1. Show that \(\mathrm { f } ( x ) = \left| x ^ { 3 } \right|\) is an even function.
  2. It is suggested that the function \(\mathrm { g } ( x ) = ( x - 1 ) ^ { 3 }\) is odd. Prove that this is false.
OCR MEI C3 Q1
1
  1. Show algebraically that the function \(\mathrm { f } ( x ) = \frac { 2 x } { 1 - x ^ { 2 } }\) is odd. Fig. 7 shows the curve \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 4\), together with the asymptote \(x = 1\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8350e810-3ceb-4876-a7a8-249e17616057-1_718_813_567_644} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
  2. Use the copy of Fig. 7 to complete the curve for \(- 4 \leqslant x \leqslant 4\).