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Representation of Functions
The function is the connection or the link between two sets and can be represented in different ways. Consider the above example of the printing machine. The function that shows the relationship between the numbers of seconds (x) and the numbers of lines printed (y). We are quite familiar with functions and now we will learn how to represent them.
Algebraic Representation
Here, we will represent the function by a simple algebraic equation which is: f(x) = y = 100 + 15(x). For different values of x, the values of y (= f(x)) change accordingly. What if, one wants to know about the numbers of lines printed by the machine in 15 seconds? Simple, Numbers of words printed = y = 100 + 15 (15) = 325.
Table Representation
In this method, we represent the relationship in the form of a table. For each value of x (input), there is one and only one value of y (output). The table representation of the problem:
Graphical Representation
Here, we will draw a graph showing the connection between the two elements of two sets say x and y such that x ∈ X and y ∈ Y. Plotting the satisfying points of x and y in the respective axes. Drawing a straight line passing through these points will represent the function in a graphical way. Graphical representation of the above problem:
Solved Examples for You
Problem: Consider an auto-driver who charges Rs. 15 for the first 7 km and subsequently charges an additional fare of Rs. 5 for each km. Find the cost one has to pay for 12 km by representing in tabular form.
Solution: Let x be the difference in distance (km) the auto ran and y be the fare of the auto for different distances (in Rs.). The equation of the given problem is: f(x) = y = 15 + 5 (x). The difference in distance travelled by auto is 12 – 7 = 5 km and we are required to find the fare for it. In tabular form, the above problem is
One has to pay Rs. 40 for travelling a distance of 12 km.
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FAQs on Representation of Functions - Algebra - Mathematics
| 1. What is the definition of a function in mathematics? |  |
Ans. In mathematics, a function is a relation between a set of inputs, called the domain, and a set of outputs, called the range, such that each input is related to exactly one output. It can be represented by an equation, a graph, or a table.
| 2. How can functions be represented graphically? | |
Ans. Functions can be represented graphically by plotting points on a coordinate plane. Each point corresponds to an input-output pair of the function. By connecting these points, a smooth curve or line can be formed, which represents the function graphically.
| 3. What is the difference between a one-to-one function and an onto function? | |
Ans. A one-to-one function is a function in which each input is related to a unique output. In other words, no two different inputs can have the same output. On the other hand, an onto function, also known as a surjective function, is a function in which every element in the range is mapped to by at least one element in the domain.
| 4. How can the domain and range of a function be determined? | |
Ans. The domain of a function is the set of all possible inputs or x-values for which the function is defined. It can be determined by looking for any restrictions or limitations on the independent variable. The range of a function is the set of all possible outputs or y-values that the function can produce. It can be determined by examining the values that the dependent variable can take.
| 5. What are some real-life examples of functions? | |
Ans. Functions are used to model various real-life situations. Some examples include the distance traveled by a car over time, the temperature of a room as a function of time, the height of a ball thrown in the air as a function of time, the sales of a product as a function of price, and the population growth of a city as a function of time.
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