Discovering the Characteristics of Hyperbolic Functions
Discovering the Characteristics of Hyperbolic Functions
The standard form of a hyperbola is the equation \(y=\dfrac{a}{x}+q\).
Domain and range
For \(y = \dfrac{a}{x} + q\), the function is undefined for \(x=0\). The domain is therefore \(\{x:x\in \mathbb{R}, x\ne 0\}\).
We see that \(y = \dfrac{a}{x} + q\) can be rewritten as:
\begin{align*} y& = \cfrac{a}{x}+q \\ y-q& = \cfrac{a}{x} \\ \text{If } x\ne 0 \text{ then: }(y-q)x& = a \\ x& = \cfrac{a}{y-q} \end{align*}This shows that the function is undefined only at \(y = q\).
Therefore the range is \(\{f(x):f(x)\in \mathbb{R}, f(x)\ne q\}\)
Example
Question
If \(g(x) = \dfrac{2}{x}+2\), determine the domain and range of the function.
Determine the domain
The domain is \(\{x:x\in \mathbb{R}, x\ne 0\}\) because \(g(x)\) is undefined only at \(x=0\).
Determine the range
We see that \(g(x)\) is undefined only at \(y = 2\). Therefore the range is \(\{g(x):g(x)\in \mathbb{R}, g(x)\ne 2\}\)
Intercepts
The \(y\)-intercept:
Every point on the \(y\)-axis has an \(x\)-coordinate of \(\text{0}\), therefore to calculate the \(y\)-intercept let \(x = 0\).
For example, the \(y\)-intercept of \(g(x) = \dfrac{2}{x}+2\) is given by setting \(x = 0\):
\begin{align*} y& = \cfrac{2}{x}+2 \\ y& = \cfrac{2}{0}+2 \end{align*}which is undefined, therefore there is no \(y\)-intercept.
The \(x\)-intercept:
Every point on the \(x\)-axis has a \(y\)-coordinate of \(\text{0}\), therefore to calculate the \(x\)-intercept, let \(y = 0\).
For example, the \(x\)-intercept of \(g(x)=\dfrac{2}{x}+2\) is given by setting \(y = 0\):
\begin{align*} y& = \cfrac{2}{x}+2 \\ 0& = \cfrac{2}{x}+2 \\ \cfrac{2}{x}& = -2 \\ x& = \cfrac{2}{-2} \\ & = -1 \end{align*}This gives the point \((-1;0)\).
Asymptotes
There are two asymptotes for functions of the form \(y = \dfrac{a}{x} + q\).
The horizontal asymptote is the line \(y = q\) and the vertical asymptote is always the \(y\)-axis, the line \(x = 0\).
Axes of symmetry
There are two lines about which a hyperbola is symmetrical: \(y = x + q\) and \(y = -x + q\).
This lesson is part of:
Functions I