IB Mathematics AA HL Flashcards- Odd & even functions

ODD/EVEN FUNCTIONS
– Even functions are functions that are symmetrical with respect to the y-axis (reflection), i.e.:
\( f(-x) = f(x) \).

[q]– Odd functions are functions that are symmetrical 180° about the origin (rotational), i.e.:
\( f(-x) = -f(x) \).

Examples:

– Odd:
\( y = x^3 \), \( y = \sin x \)

– Even:
\( y = x^2 \), \( y = x^4 \), \( y = \cos x \)

NOTE:
\( y = x^n \) is odd if \( n \) is odd, and even if \( n \) is even.

[a]

SELF-INVERSE
A self-inverse function is one that is an inverse of itself, i.e.:
If \( f(x) = f^{-1}(x) \) or \( f(f(x)) = x \).

[q]

DOMAIN RESTRICTION
Reminder: If a function is ‘one-to-many,’ then it is not technically a function. Given that an inverse essentially switches \( x \& y \), it also can switch a valid many-to-one function (e.g. \( y = x^2 \)) to an invalid function (e.g. \( y = \pm \sqrt{x} \)). You may still find the inverse, but you can restrict the domain to force it to be one-to-one. For example, \( f^{-1}(x) = \sqrt{x} \), but \( x \geq 0 \) only.

– Domain:
\( x \geq -4 \).

[x] Exit text