#### Chapter 6 Elementary Functions

Section 6.2 Linear Functions and Polynomials

# 6.2.8 Hyperbolas

We consider functions which have a reciprocal relation in their mapping rule. For the determination of the maximum domain of such a function, note that the denominator must be non-zero.
A few examples of reciprocal functions are listed below; these are reciprocals of monomials, and they are also called functions of hyperbolic type.

${f}_{1}:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ\setminus \left\{0\right\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{1}{x} ,\hfill \end{array}$

${f}_{2}:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ\setminus \left\{0\right\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{1}{{x}^{2}} ,\hfill \end{array}$

${f}_{3}:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ\setminus \left\{0\right\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{1}{{x}^{3}} ,\hfill \end{array}$

etc. Their graphs are as follows.

In particular, the graph of the function

${f}_{1}:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ\setminus \left\{0\right\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{1}{x}\hfill \end{array}$

is called the hyperbola.
Generally, for the reciprocal of an arbitrary monomial of degree $n\in ℕ$ a corresponding function of hyperbolic type can be specified.

${f}_{n}:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ\setminus \left\{0\right\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{1}{{x}^{n}} .\hfill \end{array}$

##### Exercise 6.2.14
What is the range ${W}_{{f}_{n}}$ of the function ${f}_{n}$ for even or odd $n\in ℕ$?

Further examples for functions of hyperbolic type were already considered in Example 6.1.12 and in Exercise 6.1.13 in Section 6.1.3.