In this article, we will define and elaborate on The concept of function was brought to light by mathematicians in the 17th century. In 1637, a mathematician and the first modern philosopher, Rene Descartes, talked about many mathematical relationships in his book Geometry. Still, the term “function” was officially first used by German mathematician Gottfried Wilhelm Leibniz after about fifty years. He invented a notation y = x to denote a function, dy/dx, to denote a function’s derivative. The notation y = f (x) was introduced by a Swiss mathematician Leonhard Euler in 1734. Let’s now review some key concepts as used in functions and relations.
For example, {a, b, c, …, x, y, z} is a set of alphabet letters. {…, −4, −2, 0, 2, 4, …} is a set of even numbers. {2, 3, 5, 7, 11, 13, 17, …} is a set of prime numbers Two sets are said to be equal; they contain the same members. Consider two sets, A = {1, 2, 3} and B = {3, 1, 2}. Regardless of the members’ position in sets A and B, the two sets are equal because they contain similar members. **What are ordered-pair numbers?**
A domain is a The range of a function is a collection of all output or second values. Output values are ‘y’ values of a function. In mathematics,
## Types of FunctionsFunctions can be classified in terms of relations as follows: - Injective or one-to-one function: The injective function f: P → Q implies that there is a distinct element of Q for each element of P.
- Many to one
**:**The many to one function maps two or more P’s elements to the same element of set Q. - The Surjective or onto function: This is a function for which every element of set Q there is a pre-image in set P
- Bijective function.
The common functions in algebra include: - Linear Function
- Inverse Functions
- Constant Function
- Identity Function
- Absolute Value Function
## How to Determine if a Relation is a Function?We can check if a relation is a function either graphically or by following the steps below. - Examine the x or input values.
- Examine also the y or output values.
- If all the input values are different, then the relation becomes a function, and if the values are repeated, the relation is not a function.
Identify the range and domain the relation below: {(-2, 3), {4, 5), (6, -5), (-2, 3)} Solution Since the x values are the domain, the answer is, therefore, ⟹ {-2, 4, 6} The range is {-5, 3, 5}.
Check whether the following relation is a function: B = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)} Solution B = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)} Though a relation is not classified as a function if there is a repetition of x – values, this problem is a bit tricky because x values are repeated with their corresponding y-values.
Determine the domain and range of the following function: Z = {(1, 120), (2, 100), (3, 150), (4, 130)}. Solution Domain of z = {1, 2, 3, 4 and the range is {120, 100, 150, 130}
Check if the following ordered pairs are functions: - W= {(1, 2), (2, 3), (3, 4), (4, 5)
- Y = {(1, 6), (2, 5), (1, 9), (4, 3)}
Solution - All the first values in W = {(1, 2), (2, 3), (3, 4), (4, 5)} are not repeated, therefore, this is a function.
- Y = {(1, 6), (2, 5), (1, 9), (4, 3)} is not a function because, the first value 1 has been repeated twice.
Determine whether the following ordered pairs of numbers are a function. R = (1,1); (2,2); (3,1); (4,2); (5,1); (6,7) Solution There is no repetition of x values in the given set of ordered pairs of numbers. Therefore, R = (1,1); (2,2); (3,1); (4,2); (5,1); (6,7) is a function.
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A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. So, according definition of function, given relations with domain $A = \{0, 2, 4, 6\}$, $(a) \{(6, 3), (2, 1), (0, 3), (4, 5)\} = \{ (0, 3),(2, 1), (4, 5),(6, 3)\}$ is function , since it defines for every element of function $(i.e.\{0, 2, 4, 6\})$ and has exactly one output for each element. $(b) \{(2, 3), (4, 7), (0, 1), (6, 5)\} = \{(0, 1),(2, 3), (4, 7), (6, 5)\}$ is function , since it defines for every element of function $(i.e.\{0, 2, 4, 6\})$ and has exactly one output for each element. $(c) \{(2, 1), (4, 5), (6, 3)\}$ is not a function, since it defines on element $0$ of element of function $(i.e.\{0, 2, 4, 6\})$. $(d) \{(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)\} = \{(0, 3),(0, 7),(2, 5),(4, 1),(6, 1) \}$ is not a function, since it not has exactly one output of element of function $(i.e.\{0, 2, 4, 6\})$. Element $0$ have two output $3$ and $7$. Reference@Wikipedia. |