Understanding Functions - MathsTips.com

Function Definition

Let $X$ and $Y$ be two non-empty subsets of the set of real variables $\mathbb{R}$ . If there exists a definite rule $f$ which associates each element of $x$ of $X$ to a unique element $y$ of $Y$ , then this rule $f$ is called a real valued function on the set of real variables $\mathbb{R}$ and is denoted by the symbol $f: X \rightarrow Y$ . The element in $y$ $Y$ corresponding to the element $x$ in $X$ obtained by the rule $f$ is denoted by $f(x)$ . Thus, if the rule $f$ determines $y \in Y$ when $x \in X$ , then we have, $y=f(x)$ . $f$ represents a mathematical relation which determines uniquely $y \in Y$ , when $x \in X$ .

The set of all real values over which $x$ varies is called the domain of definition of the function. The domain of definition of $f$ is $D_{f}=\left\{x:x \in \mathbb{R} \: and \: f(x) \in \mathbb{R}\right\}$ .

Again, the set of all real values of $y$ or $f(x)$ , for all $x \in D_{f}$ , is called the range of the function. Thus, range of $f$ is $R_{f}=\left\{f(x):x \in D_{f}\right\}$ .

If $y=f(x)$ is a real valued function of $x$ , then the variable $x$ to which arbitrarily different values of the domain of definition are assigned is called the independent variable, while the variable $y$ which assumes values of the range of the function is called the dependent variable.

Example:

Suppose $x$ and $y$ are two real variables and they are connected by the mathematical equation, $y=4x-5$ . then, $y=f(x)=4x-5$ will represent a real function as the relation assigns a unique value of $y=f(x)$ , for every real value of $x$ . For example, $y=4.1-5=-1$ when $x=1$ . Again, when $x=2.5, y=4 \times 2.5 -5=5$ . Clearly, the domain of definition of the function is $-\infty < x < \infty$ and its range is $-\infty < y < \infty$ .

Single-valued and Multiple-valued Functions:

If the law or rule of association between two real variables $x$ and $y$ is such that each value of x corresponds to a unique value of $y$ , then $y$ is called a single-valued function of x. However, if the law of association between them determines more than one value of $y$ for each given value of $x$ within its domain, then $y$ is called a multiple-valued function of $x$ .

For example, the function $y=8x^2$ defines y as a multiple-valued(or double-valued) function of $x$ since every positive real value of $x$ corresponds to two real values of $y$ .

A multiple-valued function is usually expressed as two or more single-valued functions by imposing conditions on the dependent variable. Thus, the multiple-valued function $y^2=4x$ can be broken into two single-valued functions, viz., $y=2\sqrt{x} (y \geq 0)$ and $y=-2\sqrt{x} (y < 0)$ .

Classification of Functions

1. Algebraic Functions

An algebraic expression in a variable containing a finite number of terms is called an algebraic function of that variable. In general, there are three types of algebraic functions:

a. Polynomial Function:

If $n$ is a positive integer and $a_{0}, a_{1}, a_{2},..., a_{n}$ are real constants, then the expression

$a_{0}x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+...+a_{n-1}x+a_{n}$

is called a polynomial function in $x$ of degree $n$ and is denoted by $P(x)$ . Thus,

$P(x)=a_{0}x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+...+a_{n-1}x+a_{n}$ , where $x$ is real.

For example, $3x^4-2x^3+6x+5, 2x^2+3x-4, 3x-2 are polynomial functions in$ latex x $ of degree 4, 3 and 1 respectively.

b. Rational Function:

The ration of two polynomial functions is called a rational function and is denoted by $R(x)$ . If $P(x)$ and $Q(x)$ are two polynomial functions, then,

$R(x)=\dfrac{P(x)}{Q(x)}$ [where $x$ is real and $Q(x) \neq 0$ ]

represents a rational function.

Each of the functions $\dfrac{3x^4-2x^3+6x+5}{3x-2}, \dfrac{ax+b}{px+q} [x \neq -\dfrac{q}{p}]$ represents a rational function.

c. Irrational Function:

An algebraic function which is not rational is called an irrational function.

Each of the functions $\sqrt[3]{x}, \sqrt{}2x^3+5$ is an irrational function.

2. Non-algebraic or Transcendental Functions

Functions which are not algebraic are called non-algebraic or transcendental functions. A few important non-algebraic functions are as follows:

a. Exponential Functions:

If $x$ is a real variable then the functions $e^x (2<e<3)$ and $a^x (a>0, a \neq 0)$ are called exponential functions.

b. Logarithmic Functions:

If $x$ is a real variable, then each of the functions $\log_{e}{x} (2<e<3; x>0), \log_{a}{x} (a>0, a \neq 1, x>0), \log_{a}{1+x} (a>0, a \neq 1, x>-1)$ is called a logarithmic function.

c. Trigonometrical Functions:

The six functions $\sin{x}, \cos{x}, \tan{x}, \sec{x}, \mbox{cosec}x, \cot{x}$ where the angle $x$ is measured in radian, are called trigonometrical functions.

d. Inverse Circular Functions:

Th six functions $\sin^{-1}x (-1 \leq x \leq 1), \cos^{-1}x (-1 \leq x \leq 1)$ , $\tan^{-1}x, \cot^{-1}x$ , $\sec^{-1}x (x \geq 1 \: or \: x \leq -1)$ , $latex \csc^{-1}{x} (x \geq 1 \: or \: x \leq -1) $ are called inverse circular functions.

3. Explicit and Implicit Functions

If the dependent variable $y$ can be expressed directly in terms of the independent variable $x$ , then the function is said to be explicit. For example, $y=2x^3-3x^2+6, f(x)=e^x+x^2$ are explicit functions. In other words, $y$ is said to be an explicit function of $x$ if it can be expressed as $y=f(x)$ .

If the dependent variable $y$ cannot be expressed directly in terms of the independent variable $x$ , then the function is said to be implicit. For example, $ax^2+2hxy+by^2=c, e^{xy}+\log{x}=5$ are implicit functions. In other words, $y$ is said to be an implicit function of $x$ if it is given in the form $f(x,y)=0$ .

Sometimes, an implicit function can be reduced to an explicit function. For example, the implicit function $3x^2-5xy+6x-8=0$ may be written as,

$y=\dfrac{3x^2+6x-8}{5x} (x \neq 0)$

which is the explicit form of the given function.However, the implicit function $e^{xy}-x^2y^2+\log{(x+y)}=0$ is not reducible to explicit form.

Parametric Form of Function

In parametric form of a function both the independent and the dependent variables are expressed in terms of a third variable. If the independent variable $x$ and the dependent variable $y$ of the function $y=f(x)$ are expressed as functions of a third variable $t$ , i.e., if $x=\phi(t), y=\psi(t)$ , then these two relations represent the parametric form of $y=f(x)$ and $t$ is called the parameter. The functional relation $y=f(x)$ can be obtained by elimination of $t$ from these two relations.

Composite Functions or Functions of Function

Let us consider the following functions:

i) $f(x)=e^{ax^2+bx+c}$

ii) $\phi(x)=\log{x}+\sqrt{x^2+a^2}$

In (i), $f(x)$ is the exponential function of $u(=ax^2+bx+c)$ and $u=ax^2+bx+c$ which is a rational function of $x$ .

Similarly, in (ii) if we assume $v=x+\sqrt{x^2+a^2}$ , then $\phi(x)=\log{v}$ .

Clearly, $\phi(x)$ is the logarithmic function of $v$ and $v$ is an irrational function of $x$ .

Hence, both $f(x)$ and $\phi(x)$ may be considered functions of another function.

Such functions like $f(x)$ and $\phi(x)$ are called composite functions.

Exercise

Does the equation $y=2x-9$ represent $y$ as a function of $x$ ? If so, find the domain of definition and range of the function.
If , prove that:
1. $f(x+2)=25f(x)$
2. $f(x+y)=f(x).f(y)$
If $f(x)=\dfrac{x-1}{x+1}$ , prove that, $\dfrac{f(x)-f(y)}{1+f(x).f(y)}=\dfrac{x-y}{1+xy}$
Find the domain of:
1. $f(x)=\dfrac{1}{\sqrt{(1-x)(x-2)}}$
2. $f(x)=\sqrt{2+x-x^2}$
Find the range of:
1. $y=\sqrt{4-x^2} [-2 \leq x \leq +2]$
2. $y=\sin{x} [0 \leq x \leq \pi]$
If, $f(x)=e^{x+a}, g(x)=x^b, h(x)=e^{b^2}x$ , prove that, $\dfrac{g\left\{f(x)\right\}}{h(x)}=e^{ab^2}$
If $f(x)=\log{x+\sqrt{1+x^2}}$ , show that, $f(-x)=f(x)$
If $f(x)=\dfrac{x-1}{x+1}$ prove that, $\dfrac{f(a)-f(b)}{1+f(a).f(b)}=\dfrac{a-b}{1+ab}$
Find the value of for which the following functions are undefined:
1. $\dfrac{x}{x+2}$
2. $\sqrt{4x-4x^2-1}$
If $e^y-e^{-y}=2x$ express $y$ as an explicit function of $x$ .