Chapter 6 Elementary Functions

Section 6.2 Linear Functions and Polynomials

6.2.2 Constant Functions and the Identity

so-called constant functions assign to every number in the domain $ℝ$ exactly the same constant number in the target set $ℝ$, e.g. the constant number $2$, in the following way:

${\displaystyle f:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & 2\hspace{0.5em}.\hfill \end{array}}$
               
Here, we then have $f(x)=2$ for all $x\in ℝ$. Hence, the range of this function $f$ consists only of the set $W_f=\{2\}\subset ℝ$.
The identity function on $ℝ$ is the function that assigns each real number to itself. This is written as follows:

${\displaystyle \mathrm{Id}:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & x\hspace{0.5em}.\hfill \end{array}}$
               
Here, we then have $\mathrm{Id}(x)=x$ for all $x\in ℝ$. Hence, the range of $\mathrm{Id}$ is the set of real numbers ($W_{\mathrm{Id}}=ℝ$). Furthermore, the identity function is (obviously) a strictly increasing function.