cadaVR anatomy by Mozaik Education

Definition and graph of a function, simple properties

Functions are one of the central, unifying concepts in mathematics. One can meet functions everywhere from geometric transformations to the concept of length, area, volume, from combinatorics to probability theory.

DEFINITION: The nonempty sets \latex{A} and \latex{B} with a given correspondence which maps (associates) every element of \latex{A} to some element of \latex{B} is called a function. A function is the triple \latex{A, B} and the said correspondence, and it is denoted by a letter (for example by \latex{f}).

The set \latex{A} is the domain of the function \latex{f}, the set \latex{B} is the codomain of the function \latex{f}. The set of those elements in \latex{B} which is the map of some element in \latex{A} is called the range of \latex{f} and it is usually denoted by \latex{f(A)}.

If \latex{x\in A}, then the element in \latex{B} to which \latex{x} is mapped to is denoted by \latex{f(x)} and it is called the value of \latex{f} at \latex{x} or the image of \latex{x} under \latex{f}. It is common to use the following notation: \latex{A = D_f} and \latex{f(A) = R_f}. The function \latex{f} can be written shortly as \latex{f: A \rightarrow B}.

Examples for functions

Let \latex{A} be the set of squares in the plane, \latex{B} the set of real numbers and map every square to its area. This way we defined a function.
Let \latex{A} and \latex{B} both be the set of points of the plane and \latex{\vec{v}} is a given vector in the plane. Map every \latex{P\in A} to the \latex{Q\in B} which is obtained by translating \latex{P} by \latex{\vec{v}}.
Let \latex{A = \left\{1, 2, …, n\right\} = B} and let \latex{i_1, i_2, …, i_n} a permutation of the numbers \latex{1, 2, …, n}. Associate \latex{i_k} with every \latex{k} (where \latex{k = 1, 2, …, n)}. Thus the given permutation can be characterized by (identified with) a function whose domain and image are both \latex{A} and the correspondence between the sets is one-to-one. We can conclude from our previous studies that the number of such functions is \latex{n!}.
Let \latex{A} be the set of possible outcomes of rolling a regular die and \latex{B} the set of real numbers from the closed interval \latex{[0;1]}. Map every element in \latex{A} to \latex{\frac{1}{6}} (the probability of the corresponding event). The resulting type of functions are called random variables.
Let \latex{A} and \latex{B} both be the set of real numbers. Map every number to its integer part. Denote the resulting function by \latex{f}. \latex{f: \R\rightarrow\R, f(x) = [x]}. The range of \latex{f} is \latex{\Z}, the set of integers.
Let \latex{A = [0, 1]}, \latex{B = \R} and denote the following function by \latex{g: A \rightarrow \R, g(x) = [x]}. The range of \latex{g} is the two-element set \latex{\left\{0;1\right\} }.

The functions \latex{f} and \latex{g} from examples \latex{5} and \latex{6} are different although there is a close connection between them.

*DEFINITION: If \latex{f: A_1 \rightarrow B} and \latex{g: A_2 \rightarrow B} are two functions for which \latex{A_2 \subseteq A_1} and for every \latex{x_2 \in A_2} it follows that \latex{f(x_2) = g(x_2)}, then \latex{f} is the extension of \latex{g} or \latex{g} is the restriction of \latex{f}.

◆ ◆ ◆

It is worth noting the following special types of functions which often play an important role.
If the function \latex{f: A \rightarrow B} satisfies that for every \latex{b \in B} there exists an \latex{a \in A} such that \latex{f(a) = b} then \latex{f} is surjective.
If the function \latex{f: A \rightarrow B} satisfies that for every \latex{a_1 \neq a_2}, \latex{a_1}, \latex{a_2 \in A} it follows that \latex{f(a_1)\neq f(a_2)} then \latex{f} is injective.
If a function \latex{f} is both injective and surjective then it is called a bijective function. A bijective function is a one-to-one correspondence between the sets \latex{A} and \latex{B}. If \latex{A = B}, then a bijective function is often called a permutation of \latex{A}.
For example the function

\latex{f: \R \rightarrow \R _0^+, f(x) = x^2} is surjective (but not injective);
\latex{g: \R_0^+ \rightarrow \R}, \latex{g(x) = x^2} is injective (but not surjective);
\latex{h: \R_0^+ \rightarrow \R_0^+}, \latex{h(x) = x^2} is bijective.

◆ ◆ ◆

In the following, unless otherwise stated, we will work with functions which map a subset of the real numbers to a subset of the real numbers, that is, functions \latex{f: A \rightarrow B} for which \latex{A, B\subseteq R}.

DEFINITION: : If the function \latex{f: A \rightarrow B} is given, then the point set

\latex{\left\{(x; f(x))\mid x\in A\right\} }

is called the graph of the function.

For example the graph of the function \latex{f: [-1; 1] \rightarrow \R, f(x) = x^3} can be seen in Figure 1.

The graph of the function \latex{g: [-2; 2] \rightarrow \R, g(x) = x - [x]} can be seen in Figure 2.

Exercises

{{exercise_number}}. Draw the graphs of the following functions.

\latex{f: [–2\pi; 2\pi]\rightarrow\R, f(x) = \sin x};

\latex{g: [0,1; 10] \rightarrow \R, g(x) = \log x};

\latex{h: \R\setminus\left]-1; 1\right[\rightarrow \R, h(x)=\frac{1}{x}};

\latex{j: [-\frac{\pi}{4}; \frac{\pi}{4} ] \rightarrow \R, j(x) = \tan x};

\latex{k:\left[1; 9\right]\rightarrow \R, k(x)=\sqrt{x}};

\latex{l: \left[-1; 1\right]\rightarrow \R, l(x) = \begin{cases} 1, \text {if }x \text{ is rational }, \\ 0, \text {if } x \text{ is irrational.} \end{cases}}

{{exercise_number}}. Decide which of the following functions are injective, surjective, bijective or none of the previous.

\latex{f: [-\frac{\pi}{2}; \frac{\pi}{2}] \rightarrow \R, f(x) = \sin x};

\latex{g: [-3; 3] \rightarrow \R, g(x) = \mid x \mid};

\latex{h: \R \rightarrow \R , h(x)=\frac{2x}{1+x^2}};

\latex{j: \R\rightarrow \Z, j(x) = [x]};

\latex{k:\R \rightarrow\R, k(x)=-2x +1};

\latex{l: \N^+ \rightarrow \R, l(x)= x}.

Mathematics 12.

-