Functions Part One
Functions Representing Relationships
A function can be said to represent a relationship between two quantities. How change in one quantity will affect the other. A lot of time these relationships are represented in equations with two variables. Both of them will change as a result of change in one another. An example equation would be how the time period of a pendulum is affected by the length of its string represented by:
T is the time period, l is the length of the string and g the gravity is a constant of 9.8 m/s². As l would change T would also change. This formula can be used to calculate the value of T for any value of l.
we can view l as the independent variable x and T as the dependent variable y. if a specific value of x can determine a singe value of y then we can say that y is a function of x or y = f(x) (y equals f of x). The letter f can be replaced with anything, it is the name of the function. Thus writing the time period equation as a function will yield the following:
There are many other relationships in mathematics which can be represented as functions.
Since the function has two variables that change in a fixed pattern (depending on the equation) a useful way for representing functions could be on the Cartesian Plane.
Example, y = x². There are infinite sets of ordered pairs in the form (x, y) for the equation y = x².
The largest possible set of values for the independent variable is called the domain of the function, the largest possible set of values for the dependent variable is called the range of the function. The ranges are the values on which the domain values are mapped. In the context of mapping, each value in the domain is mapped on to its image in the range.
Thus we can define a function as “A correspondence between two sets A and B such that each element of A corresponds to exactly one element of B. The domain is A and the range is B”
Representing the above on a mapping diagram would give us the following:
For a expression to be classified as a function it requires to have some properties. A equation can only be called a function when any value in its domain corresponds to exactly one value in its range. That is if for one value of an independent variable there are two outcomes for the dependent variable then the equation will not be classified as a function. If we look at it from a graphical context no two ordered pairs in the form (domain, range) can have the same domain.
The above is the graph for y = x², we can see that if we draw a vertical line through the graph it only passes through a single point, this is what classifies it as a function.
This is the graph for x² + y² = 23 where when we draw a vertical line it passes through two points thus saying that y is not a function of x.
Domain and Range
The domain and range of a function can be identified based on two principles and characteristics. If there is a root present in the equation of a function then the expression inside the root cannot be smaller than zero. The domain or range can be complex numbers but for all real valued functions the above is true. The second characteristic to look for is the expression or the value in the denominator, if it is equaling to zero then that particular value in the domain is invalid.
Example, The domain of y = √(3-x)
Based on what we have learnt, the expression inside the radical cannot be less than 0. Thus the domain of the function is x ≤ 3.
In the graph of any function the following would be defined as the domain and the range:
The range of the function can be easily found through the graph for the function. All the values on the y-axis for which the graph is defined come in the parlance of the range for the function. All the values on the x-axis for which the function is defined come in the parlance of the domain for the function.
Example, find the range of the function, f(x) = 1/ √(9-x²).
Upon graphing the above function we get:
We can see that graph is defined for 3≤x≤3 and ~0.33≤y≤+∞. The latter tells us that the range of the function is y ≥~0.33 or y≥1/3.
The above lesson sums up the first part of a function which includes calculation of the domain and range and the underlying math of the function schematic.
