Function properties proof

A question is this type if and only if it asks to prove properties of functions such as showing a function is odd, even, or proving that f(x) > 0 for all x.

2 questions · Moderate -0.6

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Edexcel C3 2007 January Q2
8 marks Moderate -0.3
2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$
  1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2\).
  2. Show that \(x ^ { 2 } + x + 1 > 0\) for all values of \(x\).
  3. Show that \(\mathrm { f } ( x ) > 0\) for all values of \(x , x \neq - 2\).
OCR MEI C3 2012 June Q7
4 marks Moderate -0.8
7 You are given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are odd functions, defined for \(x \in \mathbb { R }\).
  1. Given that \(\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\), prove that \(\mathrm { s } ( x )\) is an odd function.
  2. Given that \(\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )\), determine whether \(\mathrm { p } ( x )\) is odd, even or neither.