Function composition is a concept in mathematics in which two or more functions are combined into one new function. In the composition process, the first function is applied first, then the result is used as input to the second function, and so on.
In this article you will learn about the function of composition in mathematics. Make sure to pay close attention so that you do not get confused and are able to solve typical questions from this concept.
What is the configuration function?
As mentioned earlier, this is a concept in mathematics. This concept can also be called combination process or combined function because it is the combination of several functions into a new function.
The representation of this function uses the symbol “o” which indicates that two functions are composed. In order to be able to perform the composition between two functions, namely function f and function g, there are requirements that must be met.
This condition is that there must be an intersection between the resulting region of the function f and the domain of the function g which is not an empty set. In other words, there are at least some resulting values of the function f contained in the domain of the function g.
Compositions have a wide range of applications in mathematics and other sciences. An example of its application is the calculation of currency exchange rates, where there is often a configuration of currency conversion functions.
Additionally, in data processing, combination operations can be used to combine a series of data processing operations. Additionally, in modeling complex systems, this concept can help analyze the interactions and effects of different elements of the system.
Thus, composition operations are not just a theoretical concept in mathematics, but also have great practical applications in various fields of science and everyday life.
Composition and reversal functions
Compositional operations and inverse operations are two related mathematical concepts that are often used in various mathematical applications. Below is a more detailed explanation of these two concepts:
1. Configure the job
This concept is a combination of two or more functions into one new function. The process begins by applying the first function first, then the results are used as input for the second function, and so on.
Usually, the configuration process is denoted by the symbol “o”. For example, if there are two functions f(x) and g(x), the process of constructing the two functions is (gof)(x), which means that function g is applied first and the result is used as input to function f.
In order for the function f and the function g to be combined in a composition (gof), there are conditions that must be met. This condition is that there must be an intersection between the resulting region of the function f and the domain of the function g that is not an empty set.
In other words, there are at least some resulting values of the function f contained in the domain of the function g.
2. Inverse function
An inverse function is a function that is the opposite of the original function. This means that if a function is applied to its inputs and produces a given output, the inverse function will reverse the process to find the inputs from the given output.
The inverse function can only be applied to jobs that are one-to-one and to running jobs. An odd function means that each input value has only one different output value, while an on function means that each output value must have at least one corresponding input value.
The inverse function is usually denoted by f^(-1). For example, if there is a function f(x), then the inverse function is f^(-1)(x). One interesting property of the inverse function is the fact that the inverse of the inverse is itself the original function. In other words, (f^(-1))^(-1) = f.
These two concepts have different applications in mathematics and other sciences. For example, in currency exchange rate calculations, the configuration function is used to combine a series of currency conversions.
Meanwhile, the inverse function is used to find the input value from the output given by the original function. Apart from that, these two concepts are also used in data processing and modeling of complex systems.
By understanding composition and inverse operations, you can apply these concepts to understand various phenomena and more complex mathematical calculations.
An example of the application of composition operations
Some examples of applications of compositing processes are:
1. Installation process (fog)(x)
Forming a function (fog)(x) means combining two functions, namely f(x) and g(x). To find the value of (fog)(x), we must first calculate g(x) and then enter the result into the function f(x).
In other words, the first step is to perform the operation g(x), and the result will be the input function f(x). The final result of (fog)(x) is f(g(x)).
2. Installation process (gof)(x)
The installation operations (gof)(x) have a different order. This time, we will first calculate f(x) and then use the result as input to the function g(x).
So, the first step is to calculate f(x), and then the result will be the input of the function g(x). The final result of (gof)(x) is g(f(x)).
The composition function is an important mathematical concept and is frequently used in different fields. Using this concept, you can combine two or more functions into one.
Understanding this concept will be very useful for everyday life. This is because they can be applied in everyday life, allowing us to solve more complex mathematical problems and look for relationships between different functions.
The use of synthetic operations formulas can also be found in various applications in life, such as economics, computers, etc.
Example of composition function questions
Here are some example questions from compositions that you can use for practice.
Suppose there are the following two functions:
- Fungi f(x) = 2x + 3
- Fungi g(x) = x^2
Define the configuration function (gof)(x).
Step 1: Apply the function f(x) to g(x) (gof)(x) = g(f(x))
Step 2: Plug f(x) into the function g(x) (gof)(x) = g(2x + 3)
Step 3: Apply the function g(x) to the result of step 2 (gof)(x) = (2x + 3)^2
Step 4: Simplify the fitting function (gof)(x) = 4x^2 + 12x + 9
So, the configuration of the gof function is (gof)(x) = 4x^2 + 12x + 9.
You can practice solving simple problems first to understand this mathematical concept. If you already understand basic arithmetic concepts, you can try to solve several example story problems.
The following story example questions are very easy to answer because they can be applied in everyday life. Here is an example of a configuration function story question:
Example question 1
One bookstore gives a 10% discount on every purchase of more than 5 books. The price of one book is 50,000 riyals. If the buyer buys 8 books, what is the total price paid?
First, we define the functions involved in this question:
- f(x) is the discount function, where f(x) = 0.9x because a 10% discount can be interpreted as a 10% reduction in the total price.
- g(x) is the price function per book, where g(x) = 50,000x because the price of each book is IDR 50,000.
The next step is to calculate using the composition function formula (g ◦ f)(x) or g(f(x)) by combining these functions: (g ◦ f)(x) = g(f(x)) = g ( 0.9x) = 50,000 * 0.9x = 45,000x.
So, if a buyer buys 8 books, the total price paid is (g ◦ f)(8) = 45,000 * 8 = 360,000 IDR.
Example question 2
A company that produces shirts and pants. The production cost of each shirt is IDR 100,000 and each pair of trousers is IDR 150,000. If the company produces 100 shirts and 50 pants, what are the total production costs incurred?
In this case we use the following functions:
- f(x) is the clothing production cost function, where f(x) = 100,000x because the cost of producing each shirt is IDR 100,000.
- g(x) is the pants production cost function, where g(x) = 150,000x because the cost of producing each pair of pants is IDR 150,000.
Next, we calculate using the combination function formula (f + g)(x) or f(x) + g(x) to calculate the total production costs: (f + g)(x) = f(x) + g(x) = 100,000x + 150,000x = 250,000x.
Therefore, if the company produces 100 shirts and 50 pants, the total production costs incurred are (f + g) (100 + 50) = 250,000 * 150 = 37,500,000 IDR.
Example question 3
The entrepreneur owns two types of businesses, which are a bookstore and a toy store. The income from the library is IDR 10,000,000 per month and the income from the game store is IDR 5,000,000 per month. If a businessman wants to know his total income for 6 months, what is the total income he earned?
To solve this problem, we need to calculate the total income from both types of businesses for 6 months.
The first step is to define the revenue function for the bookstore and game store:
- The library’s revenue function is f(x) = 10,000,000x, where x is the number of months and 10,000,000 is the monthly revenue from the library.
- The game store’s revenue function is g(x) = 5,000,000x, where x is the number of months and 5,000,000 is the game store’s monthly income.
Then we find the total revenue from both stores for 6 months by calculating the combination function formula (f + g)(x) or f(x) + g(x): (f + g)(x) = f(x) ) + g(x) = 10,000,000x + 5,000,000x = 15,000,000x.
Next, we replace x with the value of 6 months: (f + g)(6) = 15,000,000 * 6 = 90,000,000 IDR.
Therefore, the total income earned by entrepreneurs for 6 months is IDR 90,000,000.
By paying close attention, you can answer questions about this configuration function. Of course, before solving the problem, you need to understand the meaning and how to apply it in daily life.
This mathematical concept is easy to use in everyday life. If you can understand it well, this concept will be very useful and can make it easier for you to calculate the total of many functions.