Types of Functions
The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity. An equation will be a function if for any x in the domain of the equation, there is exactly one value of y. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain, the co-domain of the function and all the assigned values are in the Range of the function. It is important how we assign the values to the each element of the domain.
Functions are of the form y = f(x), where x is an independent variable, and y is a dependent variable. A function is defined with a relation between x and y.
For example, volume and surface area of a cone and cylinder are functions of two variables radius and height.
Volume of a cuboid is a function of three variables, length, breadth and height.
In the relation y = f(x), when x=a, the corresponding value of y is f(a).
For example, when $f(x)=x^{2}-3x+4$
then f(2) = $2^{2}$ – 3 (2) + 4 = 4 – 6 + 4 = 2
f(a) = $a^{2}$ -3a + 4
1. Single valued function: If for every value of x, there correspond only one value of y, then y is called a single valued function.
Example: f(x) = 2x + 3
when x=3, f(3) = 2(3)+3 = 6+3 = 9
when x=-3, f(-3) = 2(-3)+3 = -6 + 3 = -3
2. Many valued function: If for each value of x, there corresponds more than one value of y, then y is called many or multi valued function.
Example: if $y=\pm \sqrt{}(x^{2}-9)$
when $x=5,y=\pm \sqrt{}(5^{2}-9)= \pm \sqrt{}(25-9)=\pm \sqrt{}16=\pm 4$
Hence when x=5, the two values of y are +4 and -4.
3. Explicit function: If the functional relation between the two variables x and y is expressed in the form y = f(x), y is called an explicit function of x.
Example: y = 4x -5, y = $x^{3}$ – 3x + 9
4. Implicit function: If the relation between two variables x and y is expressed in the form f(x,y)=0, where x cannot be expressed as a function of y, or y cannot be expressed as a function of x, is called an implicit function.
Example: y = $a^{x}$ , cos x + ay = btan y
Types of functions graphs
5. Odd and Even Functions: When there is no change in the sign of f(x) when x is changed to –x, then that function is called an even function. (i.e) f(-x) = f(x)
Example : $f(x)=2x^{2}-3$
$\Rightarrow f(-x)=2(-x)^{2}-3=2x^{2}-3$
$\Rightarrow f(x)=f(-x)$
Other examples are, g(x) = cos x , h(x) = $e^{x}+e^{-x}$
The graph of an even function is such that the two ends of the graph will be directed towards the same side. We can observe this from the following graph of $y=2x^{2}-3$. The graph remain unaltered when reflected about the y-axis.
When the sign of f(x) is changed when x is changed to –x, then it is called an odd function.
(i.e) f(-x) = - f(x).
Example : f(x) = $x^{3}$+ 6x
$\Rightarrow f(-x)=(-x)^{3}+6(-x)=-x^{3}-6x=-(x^{3}+6x)=-f(x)$
$\Rightarrow f(x)$ is an odd function.
Other examples are, g(x) = tan x , h(x) =$x^{5}-3x^{3}$ + 9x + 8
The following graph shows the odd function, f(x)=$x^{3}$ , and its reflection about the y-axis, which is f(-x) = $-x^{3}$.
6. Inverse functions: From every function y = f(x), we may be able to deduce a function x=g(y), such that the composition fog = Iy , gof = Ix . Meaning the composition of the function and its inverse is an identity function. In an inverse function we may be able to express the independent variable in terms of the dependent variable.
Example : $y=f(x)=\frac{(x+1)}{x-1}$
$\Rightarrow y=\frac{x+1}{x-1}$
$\Rightarrow$ y(x-1) = x+1
$\Rightarrow$ xy – y = x + 1
$\Rightarrow$ xy – x = 1 + y
$\Rightarrow$ x(y-1) = y + 1
$\Rightarrow$ $x=\frac{(y+1)}{(y-1)}=g(y)$
Let us find (gof)(x) = g [ f(x) ]
$=g\left [ \frac{(x+1)}{(x-1)} \right ]$
$=\frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=\frac{\frac{x+1+x-1}{x-1}}{\frac{x+1-x+1}{x-1}}\frac{2x}{x-1}x\frac{x-1}{2}=x$
Similarly we can prove that (fog)(y) = y, hence g is the inverse of f and f is the inverse of g.
7. Algebraic Functions: A function which consists of finite number of terms involving powers and roots of independent variable x and the four fundamental operations of addition, subtraction, multiplication and division is called an algebraic function.
Example: $f(x)=5x^{3}-2x^{2}+5x+6,g(x)=\frac{\sqrt{(2x+4)}}{(x-1)^{2}}$
Polynomials, rational functions and irrational functions are all the examples of algebraic functions.
8. Transcendental functions : Functions which are not algebraic are called transcendental functions.
Example, f(x) = sinx, g(x) = log(x), h(x)= $e^{x}$ , k(x) = tan$^{-1}$ (x)
Trigonometric functions, Inverse trigonometric functions, exponential functions, logarithmic functions are all transcendental functions.
The graph along side
shows the graph of
y= sin(x)
9. Special Functions:
(i) Modulus functions: Modulus functions are defined as follows.
y = |x | = $\left \{ \right.$ x, if x ≥ 0
- x , if x < 0.
For example, | 3 | = 3, and | -4 |= -(-4) = 4 , since -4 < 0.
The graph of the modulus function y = |x| is shown below.
(ii) Greatest integer function: The greatest integer functions are of the form y = [x]. This is also called as the step function.
For example, $[x]=0$ if $0\leq x< 1$
$[x]=1,$ if $1\leq x< 0$
The graph of the greatest integer function will be as shown below. We observe that the function is a curve and is not continuous. Hence the function is not continuous.
(iii) Constant function: If f : R $\rightarrow$ {c}, defined by f(x) = c, for every $\chi\varepsilon R$ , then f is called a constant function of R.
For example: f(x) = 8, g(x) = -5 are all constant functions.
The graph of the functions, y=8 and y = -5 are shown below.
(iv) Identity function: The identity function maps each member of the domain onto itself. That is, if I is the identity function, then for any x in the domain, I(x) = x.
It is also denoted as y = x.
The graph of the identity function is given below.
We observe that the graph passes through the points ( -4,-4),
( -2,-2), (0,0), (2,2), (4,4) and (6,6). This is because the relation is y = x.
10. Circular functions: The sine and cosine functions of elementary trigonometry and their inverses and functions derived from them.
Example : f(x) = 2sin(x)+5cos(x)
The graph of the function y = 2sin(x)+5cos(x) is given below.
The following are some of the different types of functions:
1. Onto functions:
A function f(x) is one-to-one (or injective),if f be a function with domain D and range R. A function g with domain R and range D is an inverse function for f if, for all x in D, y = f(x) if and only if x = g(y).
2. One-to-one functions:
A function f(x) is one-to-one (or injective) if, for every value of f, there is only one value of x that corresponds to that value of f.
Such that f(x) = x + 3 is is one-to-one, because, for every possible value of f(x), there is exactly one corresponding value of x.
3. Identity functions:
A polynomial of the first degree,represented as f(x) = x,
example, values of f(x) = x, at x = 1,2 are f(1) = 1 and f(2) = 2
4. Constant functions:
It is a polynomial of the zeroth degree where f(x) = cx0 = c(1) = c.It disregards the input and the result is always c. Its graph is a horizontal line.
For example f(x) = 2, whatever the value of x result is always 2.
5. Linear functions:
It is a polynomial of the first degree, the input should be multiplied by m and it adds to c. It is represented as f(x) = m x + c such as f(x) = 2x + 1 at x = 1.
f(1) = 2 . 1 + 1 = 3 that is f(1) = 3
6. Trigonometrical functions:
Trigonometric functions are often useful in modeling cyclical trends such as the seasonal variation of demand for certain items, or the cyclical nature of recession and prosperity.There are six trigonometric functions:sin(x),cos(x),tan(x),csc(x),sec(x) and cot(x)
7. Analytic functions:
All polynomials and all power series in the interior of their circle of convergence are analytic functions. Arithmetic operations of analytic function are differentiated according to the elementary rules of the calculation, and, hence analytic function.
8. Differentiable functions:
A differentiable function is a function whose derivative exists at each point in its domain. If x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f '(x0) is defined.The graph of differential functions are always smooth.
9. Smooth functions:
A smooth function is a function that has continuous derivatives over some domain or we can say that, it is a function on a Cartesian space Rn with values in the real line R if its derivatives exist at all points
10. Entire functions:
A function can be expressed as a power series with infinite radius of convergence is called an entire function.Thus exp, sin, cos, $\sinh$ and cosh are entire functions. The sum and the product of the entire function is entire. It is not true in case of a quotient.
for example, f(x) = 2 divided by g(x) = 2 - x is not entire.
11.Even and odd functions:
A function for every x in the domain of f is an even function if f(-x) = f(x) and odd function if f(-x) = - f(x)
for example, f(x) = x2 is an even function because f(-x) = (-x)2 = x2 = f(x)
and f(x) = x is an even function because f(x) = (-x) = -x = - f(x)
12. Curves functions:
A space curve C is the set of all ordered triples (f(t),g(t),h(t)) together with their
definition parametric equations
x= f(t) y = g(t) and z = h(t)
where f, g, h are continuous functions of t on an interval I.
13.Composite functions
There is one particular way to combine functions.The value of a function f depends upon the value of another variable x and that variable could be equal to another function g, so its value depends on the value of a third variable. If this is the case, then the first variable which is a function h, is called as the composition of two functions(f and g). It denoted as f o g = (f o g) x = f(g(x))
For example, let f(x) = x+1 and g(x) = 2x
then h(x) = f(g(x)) = f(2x) = 2x + 1.
14.Inverse functions:
Function g(x) is inverse function of f(x) if, for all x,
g(f(x)) = f(g(x)) = x
A function f(x) has an inverse function if and only if f(x) is one-to-one.
For example, the inverse of f(x) = x + 1 is g(x) = x - 1
15. Monotonic functions:
A monotonic function is a function that preserves the given order.These are the functions that tend to move in only one direction as x increases.
A monotonic increasing function always increases as x increases,
that is f(a) > f(b) for all a>b. A monotonic decreasing function
always decreases as x increases, that is . f(a) < f(b) for all a>b
16. Periodic functions:
Functions which repeat after a same period are called as a periodic function, such as trigonometric functions like sine and cosine are periodic functions with the period 2$\prod $.
17.Domain and range functions.
The Domain of a function is the set of all points over which it is defined. And the Range of a function is the set of all values which it attains.
for example, domain and range of function f(x) = x.
Domain of f(x) = R and Range of f(x) = R
18.Quadratic functions
A polynomial of the second degree, represented as
f(x) = ax2 + bx + c, a $\neq$ 0.
where a, b, c are constant and x is a variable.
example, value of f(x) = 2x2 + x - 1 at x = 2.
Put x = 2, f(2) = 2.22 + 2 - 1 = 9
Problem 1: Find the domain of the function f defined by f(x) = $\frac{-1}{x + 3}$
Solution: The domain of function f is the set of all values of x in the interval ( -$\infty$ , -3) U (-3 ,$\infty$ )
Problem 2: Find the domain of function f defined by f (x) = $\sqrt {-x+9}$ `
Solution: The domain of function f is the set of all values of x in the interval (-$\infty$ , 9)
Problem 3: Determine the domain of the following function:
f(x) = $\sin^{-1}\frac{x+1}{x} + \sqrt{ln\ x^{2}+ 3(\frac{x-2}{x+1}}) + \frac{1}{sinh(x+2)} + ln\sqrt{x^{2} + 12 - 2x}$
=$\sqrt{ln(x^{2}+3(x-2)/x+1}$
Solution:
1. There are no general powers to be rewritten, so we look at the function as it is given for problems. Going from the left:
2. In the arc sin there is a fraction, the denominator cannot be zero, so we get x $\neq$ 0.
3. Next we have the square root, its argument must be positive or zero:
ln`(x^(2+3)(x-2)/(x+1))` ≥ 0
4. The logarithm requires that its argument be positive:
`x^(2+3)(x-2)/x+1 ≥ 0`
5. The fraction in the logarithm needs a non-zero denominator: (x + 1) 0. Thus we get x -1.
6. The third term needs a non-zero denominator: sinh(x + 2) ≠ 0.
7. Finally we are getting to the logarithm on the right, its argument must be positive:
`sqrt(x^2+12-2x)` > 0
Then we have a root in logarithm, but its argument x2 + 12 is always at least 12, so surely we never get negative numbers and no restriction follows. Since polynomials and sinh accept any argument, there are no other conditions to consider.
