Composition of Functions

Understanding the concept of composition of functions is an important part of mathematics that helps us solve complex problems easily. A function, as you might already know, is like a machine that takes input, does something with it, and then gives an output. When we create functions, we are putting these machines together in sequence. In simple terms, one function takes the output of another as its input. Let’s dig deeper and see how this works.

What are the tasks?

Before exploring compositions, it's important to understand what functions themselves are. A function is a rule that assigns exactly one output to each input. You can think of it as a method of relating numbers or objects from one set to numbers or objects in another set.

f(x) = x + 2

In the above equation, f is a function where we input x, and it outputs x + 2 For example, if we input 3 in place of x, the output would be 5, because 3 + 2 = 5.

What is function composition?

Function composition is essentially combining two functions such that the output of the first function becomes the input for the second function. If you have two functions, f and g, the composition of these functions is represented as (f ◦ g)(x). This expression is read as "f is composed of g of x" and means f(g(x)).

Let's look at another simple task for understanding purposes:

g(x) = 2x

So if we have a function f(x) = x + 2 and another function g(x) = 2x, then composing these functions involves taking g(x) as an input to f.

Thus: (f ◦ g)(x) = f(g(x))

Substituting g(x) into f(x), we get:

f(g(x)) = f(2x) = 2x + 2

This means that when you input a number into g, multiply it by 2, and then add 2, we have created the functions f and g.

Viewing the structure of tasks

The visual representation can be a great way to understand the concept of function composition:

Imagine two separate boxes representing functions f and g. The first box takes the input, processes it according to the function g, and passes the result to the second box, which applies the function f:

Example with specific numbers

Let's apply this to specific numbers for a clearer understanding:

• Step 1: Take an input, say x = 3.
• Step 2: Apply g(x) = 2x, which gives us g(3) = 2*3 = 6.
• Step 3: Apply f(x) = x + 2 to the result of Step 2, which gives f(6) = 6 + 2 = 8.
• Step 4: Hence, (f ◦ g)(3) = 8.

Properties of function composition

It is important to understand some key properties of function composition:

Not interchangeable

Function composition is not commutative, that is, (f ◦ g)(x) is not always equal to (g ◦ f)(x):

• For our f and g as defined earlier, (g ◦ f)(x) = g(f(x)) = g(x + 2) = 2(x + 2) = 2x + 4.
• Note that this result is different from what we calculated earlier for (f ◦ g)(x).

Bonding

Function composition is associative, which means f ◦ (g ◦ h) = (f ◦ g) ◦ h. Both expressions lead to the same function when applied to the input values.

More examples of composition

Let's look at a more specific example for practice:

Example 1: Function

Consider the tasks:

f(x) = x^2
g(x) = x + 1

The combination (f ◦ g)(x) is:

• Calculate g(x) = x +1.
• Substitute into f: f(g(x)) = (g(x))^2 = (x + 1)^2.
• Expand as follows: x^2 + 2x + 1.

Example 2: Reverse order

Calculate (g ◦ f)(x) for the above functions:

• First find f(x) = x^2.
• Substitute into g: g(f(x)) = f(x) + 1 = x^2 + 1.
• So, (g ◦ f)(x) = x^2 + 1.

Why is composition important?

The structure of functions in mathematics is important and helps connect ideas together. Here are some reasons why it is important:

• Simplification of complex problems: Simplifies a series of operations into a single action.
• Real-world applications: Used in engineering, science, and data processing to build complex models to calculate outputs from multiple inputs.
• Understanding different perspectives: Provides insight into analysing systems as a set of processes applied sequentially.

Practical applications

Consider a real-world example in a business setting. Let's say a company uses two processes for pricing. First, adding a 20% markup on cost, and then a $10 service fee to the resulting price. When using a function composition, it looks like this:

m(c) = c + 0.2c
S(m) = m + 10

The combined function for the final price P based on the cost c is:

P(c) = S(m(c)) = m(c) + 10 = 1.2c + 10

This streamlines the pricing steps, ensuring consistent results across the chain of operations.

Conclusion

The composition of functions is a powerful concept that enables complex processes to be streamlined and simplified into more manageable operations. Understanding how functions are composed provides a toolkit for solving mathematical problems and applying mathematical reasoning to real-world situations.

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