Functions

The idea of a function

Definition: Let $A$ and $B$ be two sets. By a function defined on the set $A$, with values in $B$ we mean any rule (procedure or convention) $f$ which associates to every element $a\in A$ a single element, written $f(a)$, in $B$. The set $A$ is called the domain, and the set $B$ is called the codomain of the function, or the set of values of the function.

Let $f:A\to B$ be a function. By the graph of this function we mean the subset of the Cartesian product $A\times B$ formed from the pairs $(a,f(a)), a\in A$. So .

Definition: A numerical function is a function $f:A\to B$ for which both the domain $A$ and the set of values $B$ are subsets of the set of real numbers.

Injective, surjective and bijective functions

Definition: Let $f:A\to B$ be a function. We say $f$ is an injective function (one-to-one) if for any two elements $x \text{ and } y$ of $A$, $x\neq y$, we have $f(x)\neq f(y).$ The fact that $f$ is injective can also be written:

$(\forall)x,y\in A; f(x)=f(y)\Rightarrow x=y.$

Let $y=m, m\in \mathbb{R}$ be a family of lines parallel to the $Ox$ axis. If these lines meet the graph of the function $f$ in at most one point (one or none), then $f$ is injective.

Definition: Let $f:A\to B$ be a function. We say $f$ is a surjective function (onto) if for every $b\in B$ there is at least one element $a\in A$ such that $f(a)=b$. So $f:A\to B$ is not surjective if $\exists b\in B$ we have $f(a)\neq b$, $(\forall)a\in A.$

Let $y=m, m\in \mathbb{R}$ be a family of lines parallel to the $Ox$ axis. If these lines meet the graph of the function $f$ in at least one point, then $f$ is surjective.

Definition: A function $f:A\to B$ which is both injective and surjective is called a bijective function.

Let $y=m, m\in \mathbb{R}$ be a family of lines parallel to the $Ox$ axis. If these lines meet the graph of the function $f$ in exactly one point, then $f$ is bijective.

Every continuous, strictly monotonic function is bijective.

Composition of functions

Definition: Let the functions be $f:A\to B$ and $g:B\to C$ (the codomain of $f$ coincides with the domain of $g$). Let $a\in A$; then $f(a)\in B$, so its image under $g$ exists, namely $g(f(a))\in C$. We can therefore define a function $h:A\to C$ where $h(a)=g(f(a))$ for $(\forall) a\in A$. The function $h$ defined this way is written $g\circ f$ and is called the composition of the function $g$ with the function $f$.

1. If $f:A\to B$ and $g:C\to D$ are two functions, it only makes sense to speak of the composition of $g$ with $f$ when $B=C$.

2. If $f:A\to B$ and $g:B\to A$ are two functions, both $g\circ f:A\to A$ and $f\circ g:B\to B$ make sense. In general $f\circ g\neq g\circ f$.

3. Let $f:A\to B$, $g:B\to C$ and $h:C\to D$ be three functions. Then each of the functions $h\circ(g\circ f)$, $(h\circ g)\circ f$ makes sense, and the following equality holds: $h\circ(g\circ f)=(h\circ g)\circ f$.

The inverse of a function

Definition: Let $A$ be any set. Write for the function defined by for $(\forall) a\in A$. is called the identity function of the set $A$.

Let $A$ be a set and its identity function. Then:

1. For any set $B$ and any function $f:A\to B$, we have .

2. For any set $C$ and any function $g:C\to A$, we have .

Definition: A function $f:A\to B$ is called invertible if there is a function $g:B\to A$ such that and .

The graphs of the functions $f$ and are symmetric about the first bisector of the coordinate system — the line y = x.

Theorem: A function is invertible if and only if it is bijective.

Examples of invertible functions

1. If $f:\mathbb{R}\to \mathbb{R}$ is a linear function, $f(x)=ax+b, a\neq0,$ then there is a function $g:\mathbb{R}\to \mathbb{R}$, such that $f\circ g=g\circ f=x$:

So the function $f(x)=ax+b$ is invertible.

2. Let $f:[0,\infty)\to[0,\infty)$, and ,

3. Let $f:(0,\infty)\to(0,\infty)$, have , so:

4. Let , a bijective function on this domain with inverse . The graphs of these functions are shown in the figure below, so that the symmetry about the line y = x can be seen.

5. Let , a bijective function on this domain with inverse . The graphs of these functions are shown in the adjacent figure, so that the symmetry about the line y = x can be seen.

6. Let $f:[-1,2]\to[0,4]$, defined by

The function $f$ is invertible and its inverse is:

Geometric transformations of graphs

Translating graphs along the coordinate axes. Suppose the function $y=f(x)$ and the numbers $a,b>0$. Then:

1. the graph of the function $y=f(x-a)$ is obtained by translating the graph of $y=f(x)$ by $a$ units to the right;

2. the graph of the function $y=f(x+a)$ is obtained by translating the graph of $y=f(x)$ by $a$ units to the right;

3. the graph of the function $y-b=f(x)$ or $y=f(x)+b$ is obtained by translating the graph of $y=f(x)$ by $b$ units in the positive direction of the $Oy$ axis (upwards);

4. the graph of the function $y+b=f(x)$ or $y=f(x)-b$ is obtained by translating the graph of $y=f(x)$ by $b$ units in the negative direction of the $Oy$ axis (downwards).

The graph of is obtained from the graph of by two translations.

Compressing and stretching graphs. Suppose the function $y=f(x)$ and the number $a>1$. Then:

1. the graph of the function $y=f(ax)$ is obtained by compressing the graph of $y=f(x)$ horizontally by a factor of $a$;

The graph of $g(x)=sin(2\cdot x)$ is obtained from the graph of $f(x)=sin(x)$ by a contraction along the x-axis.

2. the graph of the function is obtained by stretching the graph of $y=f(x)$ horizontally by a factor of $a$;

3. the graph of the function $y=af(x)$ is obtained by stretching the graph of $y=f(x)$ vertically by a factor of $a$;

The graphs of the functions $f(x)=sin(x)$ and

The graph of the function for the following values of a = 1, 2, 3.

4. the graph of the function is obtained by compressing the graph of $y=f(x)$ vertically by a factor of $a$;

Reflecting graphs in the coordinate axes. Suppose the function $y=f(x)$. Then:

1. the graph of the function $y=f(-x)$ is obtained by reflecting the graph of $y=f(x)$ in the $Oy$ axis;

2. the graph of the function $y=-f(x)$ is obtained by reflecting the graph of $y=f(x)$ in the $Ox$ axis.

In physics one meets the function:

$A=\text{amplitude},$ $\omega=\text{angular frequency},$ $b=\text{initial phase}.$ The fundamental period of the function is

The graph of the function $f(x)=sin(x-b)$ for