Reciprocal Functions
The reciprocal function is defined as $f(x) = \frac{1}{x}$, where $x \neq 0$. This function has several unique properties.
Graph of the Reciprocal Function
The graph of $f(x) = \frac{1}{x}$ consists of two separate branches:
- A branch in the first quadrant (where both x and y are positive)
- A branch in the third quadrant (where both x and y are negative)
The x-axis and y-axis are asymptotes for this function, which means the graph gets infinitely close to these lines but never touches them.
Self-Inverse Nature
One of the most interesting properties of the reciprocal function is that it is self-inverse. This means that if you apply the function twice, you get back to where you started.
Mathematically, this can be expressed as:
$f(f(x)) = x$
ExampleLet's verify this: $$f(x) = \frac{1}{x}$$ $$f(f(x)) = f(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x$$
This property is visually represented by the symmetry of the graph about the line $y = x$.
Simple Rational Functions
A simple rational function is of the form $f(x) = \frac{ax + b}{cx + d}$, where $a$, $b$, $c$, and $d$ are constants and $c \neq 0$.
Graph Characteristics
The graphs of rational functions can have various shapes depending on the values of $a$, $b$, $c$, and $d$. However, they all share some common features:
- Vertical asymptote
- Horizontal asymptote
- x-intercept (if it exists)
- y-intercept (if it exists)
Asymptotes
Asymptotes are lines that a graph approaches but never quite reaches.
Vertical Asymptotes
A vertical asymptote occurs where the denominator of the rational function equals zero.
