The reciprocal function is defined as $f(x) = \frac{1}{x}$, where $x \neq 0$. This function has several unique properties.
The graph of $f(x) = \frac{1}{x}$ consists of two separate branches:
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.
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:
$f(f(x)) = x$
Let's verify this: $$f(x)=\frac{1}{x},\quad f(f(x))=f\left(\frac{1}{x}\right)=\frac{1}{\frac{1}{x}}=x$$
This property is visually represented by the symmetry of the graph about the line $y = x$.
A simple rational function is of the form $f(x) = \frac{ax + b}{cx + d}$, where $a$, $b$, $c$, and $d$ are constants and $c \neq 0$.
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:
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