How to Represent Functions Using Graphs, Tables, and Mappings
A function may be a relation between two sets of variables such one variable depends on another variable. We can represent differing types of functions in several ways. Usually, functions are represented using formulas or graphs. We can represent the functions in four ways as given below:
Algebraically
Numerically (Table Representation)
Visually
Verbally (Graphical Representation)
Each representation has its own advantages and disadvantages. Let’s just look and try to understand.
Different Types of Representation of functions in Maths
An example of an easy function is f(x) = x2. In this function, the function f(x) takes the given value of “x” and squares it.
For instance, if x = 3, then f(3) = 9. A few more examples of functions are: f(x) = sin x, f(x) = x2 + 3, f(x) = 1/x, f(x) = 2x + 3, etc.
There are several types of representation of functions in maths. Some important types are:
Injective function or One to at least one of the functions: When there is mapping for a variety for every domain between two sets.
Surjective functions or Onto function: Whenever there is more than one element is mapped from the domain to range.
Polynomial function: The function which consists of polynomials.
Inverse Functions: The function which inverts another function.
These were a few examples of functions. Point should be taken that there are many other functions like into function, algebraic functions, etc.
Representation of Functions
The function is the link between the two sets and it can be represented in different ways. Consider the above example of the printing machine. The function that shows the connection between the numbers of seconds (x) and therefore the numbers of lines printed (y). We are quite conversant in functions and now we'll find out how to represent them.
Algebraic Representation of Function
It is one among the standard representations of functions. In this, functions are explicitly represented using formulas. The functions are generally denoted by small letter alphabet letters. For e.g. let us take the cube function.
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The standard letter to represent function is f. However, it can be represented by any variable. To denote the function f algebraically i.e. using the formula, we write:
f : x → x3
where x is the variable denoting the input. It can be represented by any variable.
x3 is the formula of function
f is the name of the function
Even if it is one of the easiest ways of representing a function, it is not always easy to get the formula for the function. For such cases, we use different methods of representation.
In this method, we represent the connection within the sort of a table. For each value of x (input), there's one and just one value of y (output). The table representation of the problem:
Table Representation of Function
What is the Function Table?
A function table is a table of ordered pairs that follow the relationship, or rule, of a function. To make a function table for the example, first let us figure out the rule that shows our function. We have that every fraction of each day worked gives us that fraction of \[$\] 200. Thus, if we work at some point , we get \[$\] 200, because 1 * 200 = 200. If we work for two days, we get \[$\] 400, because 2 * 200 = 400. If we have to work for 1.5 days, we get \[$\] 300 in amount, as 1.5 * 200 = 300. Are we seeing a pattern here?
To find the entire amount of cash made at this job, we multiply the amount of days we've worked by 200. Thus, our rule is that we take a worth of x (the number of days worked), and that we multiply it by 200 to urge y (the total amount of money made).
A function table is used to display the rules. In the first row for the function table, we put the values of x, and in the second row of the table, we put the corresponding values of y which is according to the function rule.
Graphical Representation of Function
Here, we'll draw a graph showing the connection between the 2 elements of two sets, say x and y such that x ∈ X and y ∈ Y. Putting up the satisfying points of x and y in their own axes. Drawing a line passing through these points will represent the function during a graphical way. Graphical representation of the above problem:
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FAQs on Representation of Functions Explained with Examples
1. What is meant by the representation of a function in mathematics?
The representation of a function refers to the various ways a mathematical function can be expressed, such as using algebraic formulas, tables, graphs, or verbal descriptions. Each method highlights different aspects of the function and helps in understanding its behavior as per the CBSE Class 12 Maths syllabus.
2. How can functions be represented algebraically, numerically, graphically, and verbally?
Functions can be expressed in four main ways:
- Algebraically: Using a formula (e.g., f(x) = x2 + 3)
- Numerically (Table): A table lists input-output pairs
- Graphically: As a curve or line on a Cartesian plane
- Verbally: Describing the relationship in words
3. Why is it important to use different representations of functions?
Using different representations makes it easier to analyse, interpret, and solve problems involving functions. For example, a graph can help identify trends, while an algebraic formula is useful for calculation. Verbal descriptions can clarify context, and tables can organise data systematically.
4. How can you determine if a given relation is a function using its graphical representation?
The vertical line test is used: if any vertical line drawn on the graph of a relation intersects it more than once, it is not a function. If every vertical line crosses the graph at most once, the relation is a function. This test is especially relevant for exam questions on identifying functions.
5. What are injective, surjective and bijective functions?
An injective function (one-to-one) maps each element of the domain to a unique element in the range. A surjective function (onto) covers all elements of the range. A bijective function is both injective and surjective, meaning there is a perfect pairing between domain and range. Understanding these types is essential for CBSE board exams.
6. Can every mathematical relationship be represented as a function? Explain with an example.
No, not every relationship is a function. For a relation to be a function, each input must be associated with exactly one output. For example, the relation y2 = x is not a function because a single value of x can correspond to more than one value of y.
7. What are some real-life applications of representing functions?
Functions model real-life situations such as predicting profit based on sales, calculating distance-time relationships, or converting temperatures. Representation helps in interpreting, forecasting, and making decisions based on the mathematical model of the situation.
8. How do you convert a table of values into a function?
To convert a table of values into a function, observe the pattern between the inputs (x) and outputs (y). Find the mathematical rule that connects them, and express it as an algebraic formula, ensuring each input has only one corresponding output.
9. What misconceptions do students often have about functions and their representations?
Common misconceptions include believing every graph is a function, confusing relations with functions, or thinking that the representation method does not affect problem-solving. It is important to understand the definition of a function and carefully check for uniqueness in outputs for each input.
10. If a function is defined verbally, how can you translate it to an algebraic or graphical form?
First, carefully read the verbal definition to identify the relationship between variables. Translate the words into a mathematical expression or equation, then plot the corresponding points on a graph to obtain the graphical representation of the function.
