Odd and Even Functions
Odd and even functions are special types of functions that exhibit particular symmetry properties.
Even Functions
An even function is symmetric about the y-axis. Mathematically, a function $f(x)$ is even if:
$$f(-x) = f(x)$$
for all $x$ in the domain of $f$.
ExampleThe function $f(x) = x^2$ is an even function because $f(-x) = (-x)^2 = x^2 = f(x)$.
Graphically, even functions can be recognized by their mirror symmetry about the y-axis.
Odd Functions
An odd function has rotational symmetry of 180° about the origin. Mathematically, a function $f(x)$ is odd if:
$$f(-x) = -f(x)$$
for all $x$ in the domain of $f$.
ExampleThe function $f(x) = x^3$ is an odd function because $f(-x) = (-x)^3 = -x^3 = -f(x)$.
Graphically, odd functions can be recognised by their rotational symmetry about the origin.
NoteNot all functions are either odd or even. For example, $f(x) = x^2 + x$ is neither odd nor even.
Periodic Functions
Periodic functions can also be classified as odd or even. A periodic function $f(x)$ with period $T$ is true when
$$f(x+T) = f(x)$$
for all $x$ defined in the function.
Inverse Functions and Domain Restriction
Finding Inverse Functions
The inverse of a function $f$, denoted as $f^{-1}$, "undoes" what $f$ does. If $f$ maps $x$ to $y$, then $f^{-1}$ maps $y$ back to $x$. Mathematically:
$$f(f^{-1}(x)) = f^{-1}(f(x)) = x$$
To find the inverse function:
- Replace $f(x)$ with $y$
