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 $f$.
The 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 $f$.
The 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.
Not 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 $y$
- Swap $x$ and $y$
- Solve for $y$
- Replace $y$ with $f^{-1}(x)$
Let's find the inverse of $f(x) = 2x + 3$:
- $y = 2x + 3$
- $x = 2y + 3$
- $x - 3 = 2y$ $\frac{x - 3}{2} = y$
- $f^{-1}(x) = \frac{x - 3}{2}$
Domain Restriction
Not all functions have inverses over their entire domain. For a function to have an inverse, it must be one-to-one (injective) and onto (surjective), i.e., bijective.
Students often forget to check if a function is bijective before finding its inverse.
To make a non-bijective function invertible, we can restrict its domain. This is particularly common with functions like $f(x) = x^2$, which is not one-to-one over its entire domain.
For $f(x) = x^2$, we can restrict the domain to $[0, \infty)$ to make it invertible. The inverse function is then $f^{-1}(x) = \sqrt{x}$, defined only for non-negative real numbers.
When finding inverses, always consider the domain and range of both the original function and its inverse.
Self-Inverse Functions
A self-inverse function is a function that is its own inverse. Mathematically, for a self-inverse function $f$:
$$f(f(x)) = x$$
or
$$f(x) = f^{-1}(x)$$
for all $x$ in the domain of $f$.
Self-inverse functions have some interesting properties:
- They are always bijective.
- Their graphs are symmetric about the line $y = x$.
The function $f(x) = \frac{1}{x}$ for $x \neq 0$ is self-inverse:
$f(f(x)) = f(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x$
Other examples of self-inverse functions include:
- $f(x) = -x$
- $f(x) = \frac{a-x}{x+1}$, where $a$ is a constant and $x \neq -1$
Self-inverse functions are relatively rare and have unique properties that make them interesting in various mathematical contexts.
