Introduction to Reciprocals
Reciprocal Functions occur when a polynomial function is divided by another polynomial function. They are expressed algebraically as:
Reciprocal functions can be represented graphically as:
The numerator of a reciprocal function is always represented as a numeric constant such as \(1\) or \(2\).
Conversely, the denominators can range from single numeric terms to non-numeric terms such as variables, such as \(n\) or \(p\), or polynomial expressions, such as \(5x^2 + 3\).
In order to determine the reciprocal (or inverse) of a function, divide the original expression by \(1\). For instance, the reciprocal of \(x - 4\) is \(\cfrac{1}{x-4}\).
Reciprocal functions with numerators different than than \(1\) can be expressed algebraically as:
Where \(k\) represents the numerator.
Reciprocal functions in Standard Form can be represented algebraically as:
Where:
- \(a\) represents a common number
- \(x\) represents the \(x\)-value (or input)
- \(h\) represents the Vertical Asymptote
- \(k\) represents the Horizontal Asymptote
Characteristics of Reciprocal Functions
Domain
The Domain of reciprocal functions is all real numbers where \(x\) cannot equal \(0\), \(\{x\in\mathbb{R}, x \ne 0\}\). If the function is in Standard form, then the domain will include all real numbers where x cannot equal \(k\), \(\{x\in\mathbb{R}, x \ne k\}\).
Range
The Range of reciprocal functions is all real numbers where \(x\) cannot equal \(0\), \(\{y\in\mathbb{R}, y \ne 0\}\). If the function is in Standard Form, then the range will include all real numbers where \(x\) cannot equal \(h\), \(\{x\in\mathbb{R}, x \ne h\}\).
Intercepts
The y-intercept can be determined by substituting \(0\) for \(x\) and solving for \(f(x)\). This can only be done if the function has an \(h\)-value. Otherwise, it will make the denominator \(0\), thereby making the entire function undefined.
The x-intercept(s) can be determined by setting the numerator equal to \(0\) and solving for \(x\).
Asymptotes
Vertical Asymptotes are vertical lines that a function cannot cross, thereby restricting its domain. In a reciprocal function, the default VA will be \(x=0\), better known as the \(y\)-axis. When the reciprocal function is in Standard Form, the VA will be \(x=h\).
Horizontal Asymptotes are horizontal lines that a function approaches as its inputs, \(x\), approaches or \(-\infty\). Unlike its vertical counterpart, a Horizontal Asymptote can be crossed. In a reciprocal function, the default HA will be \(y=0\), better known as the \(x\)-axis. When the reciprocal function is in Standard Form, the HA will be \(x=k\).
End Behaviours
The end behavior for reciprocal functions are dependent on the function’s sign and are summarized in the table below assuming that \(f(x) = \cfrac{1}{x}\):
| Sign | Left Branch | Right Branch |
|---|---|---|
| Positive | \(x \rightarrow -\infty, y \rightarrow 0\) \(x \rightarrow 0, y \rightarrow -\infty\) |
\(x \rightarrow 0, y \rightarrow +\infty\) \(x \rightarrow +\infty, y \rightarrow 0\) |
| Negative | \(x \rightarrow -\infty, y \rightarrow 0\) \(x \rightarrow 0, y \rightarrow +\infty\) |
\(x \rightarrow 0, y \rightarrow -\infty\) \(x \rightarrow +\infty, y \rightarrow 0\) |
Outlined below are the end behaviors for reciprocal functions where \(f(x) = \cfrac{a}{x - h} + k\):
| Sign | Left Branch | Right Branch |
|---|---|---|
| Positive | \(x \rightarrow -\infty, y \rightarrow k\) \(x \rightarrow h^-, y \rightarrow -\infty\) |
\(x \rightarrow h^+, y \rightarrow +\infty\) \(x \rightarrow +\infty, y \rightarrow k\) |
| Negative | \(x \rightarrow -\infty, y \rightarrow k\) \(x \rightarrow h^-, y \rightarrow +\infty\) |
\(x \rightarrow h^+, y \rightarrow -\infty\) \(x \rightarrow +\infty, y \rightarrow k\) |