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Representation of a function is a uniform depiction of multidimensional objects. A function, in mathematics, is an impression or a principle that describes a relationship between one independent variable and another variable (the dependent variable). It is an integral part of mathematics.
- Representation of a function is done with the help of formulas and graphs.
- An object is defined using single real-valued functions.
- A function has a domain, codomain, or range.
- It is most frequently denoted by letters like f, g, and h.
- Representation of a function is denoted as f with x of its domain, which is denoted by f(x).
- The concept is widely used in the field of mathematics.
- They are represented in pairs x, and f(x), which are called graphs of the function.
- Representation of a function is used in volume modelling, solid modelling and computer graphics.
Read More: First Order Differential Equation
Key Terms: Representation of a function, Function, Variable, Domain, Range Value, Set, Graph, Mappings, Polynomial Function, Input, Output
What is a Function?
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A function is outlined as a relation between a collection of inputs having one output each. It can be defined as a relationship between inputs and outputs where every input is described in terms of one output.
- Functions are also called maps or mappings.
- It is generally denoted by f(x), where x is the independent variable, and y is the dependent variable.
- The general representation of a function is y = f(x).
- Functions that involve more than two variables are known as multivariable or multivariate functions and are most common in mathematics.
For example:
- A circle with area A has a formula of A = πr2,
- With the dependent variable (the area) = A
- The function of the independent variable (the radius) = r
Function
Read More: Geometry Formula: Area, Perimeter
Types of function
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The different types of functions are as follows:
Injective function
Injective Functions are also called one-to-one functions. In these types of functions, there is mapping for a function for each domain between two sets.
Surjective Function
Surjective Functions are also called onto functions. In these functions, there is more than one element mapped from domain to range.
Read More: Hyperbolic Functions Formula
Polynomial Function
A polynomial function is defined as a function which involves a non-negative integer powers. It is represented by P(x), and the highest power of a polynomial is called a degree. The polynomial function is represented as:
P(x) = an xn + an-1 xn-1+.……….…+a2 x2 + a1 x + a0, where x > 0 or x < 0, P(x) ≈ an xn.
Inverse function
An inverse function is a function that returns the original value of the function for the given output. This function can invert another function. It is also known as anti-function. In this, the independent variable is changed with the dependent variable.
There are many other functions, like algebraic functions, math functions, etc, which are incorporated in the representation of a function.
Read More: Tangent to a Circle
Solved Examples for Function
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Given below are some example of calculating the value of x by various function.
Example 1: Consider a function which is f(x) = x2.
Ans. In this equation, function f(x) squares the given value of “x”.
If x = 2, then f(2) = 4.
Read More: Differentiation and Integration Formula
Example 2: Consider a function which is f(x) = 2x + 3.
Ans. In this equation, function f(x) is giving the various value of x.
If x = 2 then f(2) = 2 x 2 + 3
f(2) = 4 + 3
f(2) = 7
Read More: Algebra Formula
Representation of a Function
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The function is the link between two sets. Representation of a function in different ways and the relationship is symbolized as y = f(x)—which is called “f of x”—and y and x are related such that for every x, there is a unique value of y.
The four ways for the representation of a function are given below.
- Algebraic Representation
- Numeric Representation
- Verbal Representation
- Visual Representation
| Types | Description of a function |
|---|---|
| Algebraic Representation of a function | In this function, formulas and mathematical model are used to express the function. |
| Numeric Representation of a function | In this function, tables or chart are used to represent the value of function. |
| Verbal Representation of a function | In this function, values are represented verbally. |
| Visual Representation of a function | In this function, the values are represented through graphical method. |
Representation of a Function
The detailed analysis of different ways for representation of function are as follows:
Algebraic Representation of a Function
Algebraic Representation is one of the standard representations of functions. It gives the expression of a function using an equation or mathematical model.
- Functions are denoted by small alphabet letters.
- The letter to represent function is f.
- Depiction of a group of the function, f algebraically, i.e., using the formula, we get
f(x:x) → x3
Here x is the variable that indicates the input.
Read More: Introduction, Different Branches of Algebra, Equations.
Numeric Representation of a Function
In Numeric Representation of a function, the values of a function are represented using tables or values or charts.
- The table comprises of two columns; one with the dependent variable and the other with the independent variable.
- The output can only have a finite number of value.
- Hence the analysis of the function and study becomes difficult.
Solved Example For Numeric Representation of a Function
Taking function f and independent variable x. Suppose we have a function called f(x) = 2x
The tabular representation of a function is as follows:
| x | f(x) |
|---|---|
| 4 | 16 |
| -2 | -4 |
| 9 | 18 |
| -0.5 | -1 |
| 1.5 | 3 |
Verbal Representation of a Function
- Words are used to represent the function.
- An example would be – For the input x, the function gives the largest integer smaller than or equal to x i.e. floor function.
Visual Representation of a Function
- Representation of function in the form of graphs.
- An easy form of representation to understand.
- Input values are marked along the x-axis and the corresponding output value is the vertical displacement from the x-axis.
Read More: Differences Between Relation and Function
Things to Remember
- Representation of a function is the depiction of different values of a function.
- A function is a relationship between two independent and dependent variables.
- Representation of a function is divided into four categories, namely Algebraic Representation, Numeric Representation, Verbal Representation and Visual Representation.
- The function was first defined by the German mathematician Peter Dirichlet in 1837.
- A function comprises a domain and codomain for the given variable x.
- A domain is the set of inputs for which the function is defined = x value ( first value in the ordered set.)
- A codomain is the set of possible output values = y set( second value of the ordered set).
Read More: Domain and Range of Trigonometric Functions.
Sample questions -
Ques. Find x and y if
(a) ( 4x + 3,y ) = ( 3x + 5, – 2 )
(b) ( x - y ) ( x + y )= (6,10 ) (3 Marks)
Ans.
- Since (4x + 3,y ) = ( 3x + 5,- 2 )
4x+3 = 3x+ 5
Or x= 2
y= –2
- x–y = 6
x+y = 10
Therefore,
2x= 16
Or x = 8
8–y = 6
y= 2.
Ques. Find the domain for which of the function f(x)= 2x2 – 1 and g (x) = 1 – 3x are equal (2 Marks)
Ans. For f (x) = g (x)
2x2 -1 =1 - 3x
2x2 + 3x – 2 = 0
2x2 + 4x – x – x – 2 = 0
2x ( x + 2 ) – 1 (x + 2 ) = 0
(2x – 1 ) (x + 2 ) = 0
Therefore domain for which the function f(x) = g(x) if 12, – 2
Ques. If A= 2,4,6,9 and B = 4,6,18,27,54, a∈ A ,b∈ B , find the set of ordered pairs such that a is a factor of b and a< b (3 Marks)
Ans. Since, A= 2,4,6,9 and B= 4,6,18,27,54, we have to find a set of ordered pairs a and b such that a is a factor of b and a< b.
Since 2 is a factor of 4 and 2< 4
So 2 and 4 are one such ordered pair.
Similarly (2.6 ), (2, 18 ), ( 2,54 ) are other such ordered pairs. Thus the required set of ordered pairs are
(2,4 ), ( 2, 6 ), ( 2, 18 ), ( 2,54 ), ( 6,18 ), ( 6,54 ), ( 9,18 ), ( 9,27 ) ,( 9,54 ).
Ques. Is the following relation a function? Give a justification of the answer. (2 Marks)
(a) R1 = ( 2,3 ), ( 12, 0 ), (2,7 ),( -4,6).
(b) R2 = (x, IxI ) x is a real number .
Ans. Since (2,3 ) and (2,7 ) ∈ R1
R2 (2 ) =3 and R1 (2 ) =7.
So R1 (2 ) does not have a unique image. Thus R1 is not a function.
R2 = (x, IxI ) x ∈ R
For every x ∈ R there will be a unique image as IxI ∈ R
Therefore R2 is a function.
Ques. Find the domain of the function f given by f(x) = 1x2 – x – 6. (2 Marks)
Ans. Given f (x) = 1x2 – x – 6 , f is defined if x2 – x – 6 > 0
Or ([x] – 3) ([x] + 2) > 0,
x < – 2 or x > 3
x < – 2, x> / 4
Hence domain = (- ∞, -2 ) ∪ ( 4, ∞ ).
Ques. If f ( x) = x3 – 1x3, then f(x) + f (1x) is equal to. (2 Marks)
(a) 2x3
(b) 21x3
(c) 0
(d) 1
Ans. The correct answer is c.
Explaination: Since f ( x) = x3 - 1x3
= f ( 1x) = 1x3 - 11 /x3
= 1x3 - x3
Hence, f ( x) + f ( 1x ) = x3 – 1x3 + 1x3 – x3 = 0
Ques. Let f and g be two functions given by f = ( 2,4 ) ( 5,6 ) ( 8,-1 ) ( 10,-3 ), g = (2,5 ) ( 7,1 ) (8,4 ) (10,13 ) ( 11, -5 ), then the domain of f + g is. (2 Marks)
Ans. Since domain of f= Df= (2,5,8,10 )
And domain of g is Dg= ( 2,7,8,10,11 )
Therefore the domain of f+g is ( x IxI x∈ Df âÂ<Â, Dg ) = 2,8, 10.
Ques. If (x/3 + 1, y – 2/3) = (5/3, 1/3), find the values of x and y. (3 Marks)
Ans. It is given that (x/3+1,y-2/3)= (5/3,1/3)
Since the ordered pairs are equal, the corresponding elements will also be equal.
Therefore, x/3 + 1=5/3 and y-2/3=1/3
x/3 + 1= 5/3
⇒ x/3=5/3-1 y-2/3=1/3
⇒ x/3=2/3 ⇒ y=1/3+2/3
⇒ x=2 ⇒ y=1
Therefore x= 2 and y=1
Ques. Calculate the inverse function of f(x) = 4x + 5. (2 Marks)
Ans. Let f(x) = 4x + 5 = y
y = 4x + 5
y – 5 = 4x
x= (y – 5) / 4
So the value of inverse function is given as (y – 5) / 4
Ques. Find the inverse of the function f(x) = ln(x – 5). (2 Marks)
Ans. First, replace f(x) with y
So, y = ln(x – 5)
Replace the equation in exponential way , x – 5 = ey
Now, solving for x,
x = 5 + ey
Now, replace x with y and thus, f-1(x) = y = 5 + ey
Ques. Consider f(x) = 2x + 10, then calculate the value of function at x = 4 and x = 6. (2 Marks)
Ans. We have, f(x) = 2x +10
f(4) = 2 × 4 + 10
f(4) = 11
Now, let’s apply for x = 6.
f(6) = 2 × 6 + 10
f(6) = 22
Ques. For the given functions f(x) = 5x + 7 and g(x) = 3x - 2, find the value of fog(x). (3 Marks)
Ans. The given two functions are f(x) = 5x + 7 and g(x) = 2x - 1.
The function fog(x) is to be found.
fog(x) = f(g(x))
= f(3x – 2)
= 5(2x - 1) + 7
= 10x - 5 + 2
= 10x - 3
Therefore fog(x) = 10x - 3
Ques. Find the domain and the range of the real function, f(x) = 1/(x + 8). (2 Marks)
Ans. We have f(x) = 1/(x + 8)
Clearly, f is not defined for x = -8
Therefore, dom(f) = R – {-8}
Let y = f(x). Then,
y = 1/(x + 8) ⇒ x = (1/y) – 8 …….(i)
Clearly, (i) is not defined for y = 0
Therefore, range(f) = R – {0}
Ques. Let f: R →R: f(x) = x3 and g: R →R: g(x) = x + 1. Find (f + g)(x). (2 Marks)
Ans. Here, dom(f) = R = dom(g)
Therefore, dom(f) ∩ dom(g) = R
Then, (f + g)(x) = f(x) + g(x) = x3 + x + 1
Ques. Calculate the inverse function of f(x) = 3x + 26 / 4. (2 Marks)
Ans. Let f(x) = x + 26 / 4 = y
y = 3x + 26 / 4
4y = 3x + 26
4y – 26 = 3x
x= (4y – 26) / 3
So the value of inverse function is given as (4y – 26) / 3
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