4.4 Linear function

We call linear function 39 any function, for which there exists costants $a,b\in\mathbb{R}$ such that

\begin{equation}\label{eq-lin-fce}\tag{4.6} f(x) = ax + b \end{equation}

holds for every $x\in\mathbb{R}$. The graph of a linear function is a straigh line, see picture 4.3.

Figure 4.3: Graph of a linear function.

The domain of a linear function is the whole real line $\mathbb{R}$. If $a\neq 0$, then the image set of function (4.6) is given by the set of all real numbers. In the case $a = 0$ the image set of the function (4.6) is the set $H_f = \{b\}$. In short,

\begin{equation*} \begin{aligned} D_f &= \mathbb{R}, \\ H_f &= \begin{cases} \mathbb{R}, & a\neq 0, \\ \{b\}, & a = 0. \end{cases} \end{aligned} \end{equation*}

In the special case with vanishing $a$, i.e. $f(x) = b$, we speak of constant function.

The roots of a linear function are easy to find, for example the equation $ax + b = 0$ has solution $x = \frac{-b}{a}$ if $a$ is not zero. In case $a=0$ and $b$ is not vanishing, then the corresponding equation has no solution and no intersection with the $x$ axis exists. In case both $a$ and $b$ equal to zero, we have the zero fuction, whose roots are given by the set of all real numbers.

Question 4.2

At the beginning of this section it was said that the graph of each linear function is a straight line. On the contrary, is every straigh line the graph of a linear function?

Show answer

It is not. For example, any straight line parallel to the $y$ axis is not a function, it can not be expressed by the equation $y = a x + b$ with real $a,\, b$.

Question 4.3

Above examples of a linear function $f(x) = ax + b$ have been metioned, which have just one, or no intersection with the $x$ axis. Do they exhaust all the possibilities regarding the number of intersection with the $x$ axis?

Show answer

No. The linear fuction $f(x) = 0$ coincides with the $x$ axis and thus the number of intersections is infinite.

Remark 4.2 (Terminology)

In the second semester, you will study Linear Algebra ( BIE-LIN), where the term linear operator plays a central role. At this point the reader should note that the word „linear“ in linear algebra means to require the following property:

\begin{equation*} f(x + \alpha y) = f(x) + \alpha f(y) \end{equation*}

for all the vectors $x,y$ and all the numbers $\alpha$. This condition is satisfyed by the linear function introduced here only if $b = 0$, i,e. only when theirs graphs pass through the origin of the coordinates system. Our linear function $f(x) = ax + b$ for non-vaishing $b$ is called in linear algebra affine function (operator).