Composition of functions with the same domain and range

Oct 13, 2025 by windailys.com 93 views

Introduction

In mathematics, function composition and operations are essential concepts that allow us to combine and manipulate functions to create new ones. Given two functions, f(x) and g(x), we can perform various operations such as addition, subtraction, multiplication, and division to obtain new functions. In this article, we will focus on finding the sum and difference of two functions, (f+g)(x) and (f-g)(x), using the given functions f(x) = 5x - 6 and g(x) = x^2 - 5x + 6.

Step 1: Understanding Function Composition

Before we proceed with finding (f+g)(x) and (f-g)(x), let's briefly review function composition. Function composition is the process of combining two or more functions to create a new function. Given two functions f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), is defined as:

(f ∘ g)(x) = f(g(x))

In other words, we first apply the function g to x, and then apply the function f to the result.

Step 2: Finding (f+g)(x)

To find (f+g)(x), we need to add the two functions f(x) and g(x) together. This means we will combine the like terms and simplify the resulting expression.

(f+g)(x) = f(x) + g(x) = (5x - 6) + (x^2 - 5x + 6) = x^2 - 5x + 6 + 5x - 6 = x^2

Step 3: Finding (f-g)(x)

To find (f-g)(x), we need to subtract the function g(x) from f(x). This means we will combine the like terms and simplify the resulting expression.

(f-g)(x) = f(x) - g(x) = (5x - 6) - (x^2 - 5x + 6) = 5x - 6 - x^2 + 5x - 6 = -x^2 + 10x - 12

Conclusion

In this article, we have successfully found the sum and difference of two functions, (f+g)(x) and (f-g)(x), using the given functions f(x) = 5x - 6 and g(x) = x^2 - 5x + 6. We have demonstrated the importance of function composition and operations in mathematics, and have shown how to combine and manipulate functions to create new ones.

Example Problems

Given f(x) = 2x + 3 and g(x) = x^2 - 4x + 5, find (f+g)(x) and (f-g)(x).
Given f(x) = x^2 + 2x - 3 and g(x) = 3x - 2, find (f+g)(x) and (f-g)(x).

Solutions

(f+g)(x) = (2x + 3) + (x^2 - 4x + 5) = x^2 - 2x + 8 (f-g)(x) = (2x + 3) - (x^2 - 4x + 5) = -x^2 + 6x - 2
(f+g)(x) = (x^2 + 2x - 3) + (3x - 2) = x^2 + 5x - 5 (f-g)(x) = (x^2 + 2x - 3) - (3x - 2) = x^2 - x - 1

Applications

Function composition and operations have numerous applications in mathematics, science, and engineering. Some of the key applications include:

Algebra: Function composition and operations are used to solve equations and inequalities, and to manipulate expressions.
Calculus: Function composition and operations are used to find derivatives and integrals, and to solve optimization problems.
Physics: Function composition and operations are used to model real-world phenomena, such as motion and energy.
Computer Science: Function composition and operations are used to develop algorithms and software, and to solve problems in computer science.

Conclusion

Introduction

In our previous article, we explored the concept of function composition and operations, and demonstrated how to find the sum and difference of two functions, (f+g)(x) and (f-g)(x). In this article, we will answer some frequently asked questions (FAQs) related to function composition and operations.

Q1: What is function composition?

A1: Function composition is the process of combining two or more functions to create a new function. Given two functions f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), is defined as:

(f ∘ g)(x) = f(g(x))

In other words, we first apply the function g to x, and then apply the function f to the result.

Q2: What are the different types of function composition?

A2: There are several types of function composition, including:

Pointwise composition: This is the most common type of function composition, where we combine two functions pointwise.
Composition of functions with the same domain and range: This type of composition is used when the two functions have the same domain and range.
Composition of functions with different domains and ranges: This type of composition is used when the two functions have different domains and ranges.

Q3: How do I find the sum and difference of two functions?

A3: To find the sum and difference of two functions, we need to add or subtract the two functions together. This means we will combine the like terms and simplify the resulting expression.

For example, given f(x) = 5x - 6 and g(x) = x^2 - 5x + 6, we can find the sum and difference of the two functions as follows:

(f+g)(x) = f(x) + g(x) = (5x - 6) + (x^2 - 5x + 6) = x^2

(f-g)(x) = f(x) - g(x) = (5x - 6) - (x^2 - 5x + 6) = -x^2 + 10x - 12

Q4: What are some common applications of function composition and operations?

A4: Function composition and operations have numerous applications in mathematics, science, and engineering. Some of the key applications include:

Algebra: Function composition and operations are used to solve equations and inequalities, and to manipulate expressions.
Calculus: Function composition and operations are used to find derivatives and integrals, and to solve optimization problems.
Physics: Function composition and operations are used to model real-world phenomena, such as motion and energy.
Computer Science: Function composition and operations are used to develop algorithms and software, and to solve problems in computer science.

Q5: How do I determine if a function is a composition of two functions?

A5: To determine if a function is a composition of two functions, we need to check if the function can be written as a combination of two or more functions. We can do this by looking for the following patterns:

Function composition: If the function can be written as f(g(x)), where f and g are two functions, then it is a composition of two functions.
Function addition: If the function can be written as f(x) + g(x), where f and g are two functions, then it is a sum of two functions.
Function subtraction: If the function can be written as f(x) - g(x), where f and g are two functions, then it is a difference of two functions.

Conclusion

In conclusion, function composition and operations are essential concepts in mathematics that allow us to combine and manipulate functions to create new ones. By understanding and applying these concepts, we can solve a wide range of problems in mathematics, science, and engineering. We hope that this Q&A article has provided you with a better understanding of function composition and operations, and has helped you to develop your problem-solving skills.

Example Problems

Given f(x) = 2x + 3 and g(x) = x^2 - 4x + 5, find (f+g)(x) and (f-g)(x).
Given f(x) = x^2 + 2x - 3 and g(x) = 3x - 2, find (f+g)(x) and (f-g)(x).

Solutions

(f+g)(x) = (2x + 3) + (x^2 - 4x + 5) = x^2 - 2x + 8 (f-g)(x) = (2x + 3) - (x^2 - 4x + 5) = -x^2 + 6x - 2
(f+g)(x) = (x^2 + 2x - 3) + (3x - 2) = x^2 + 5x - 5 (f-g)(x) = (x^2 + 2x - 3) - (3x - 2) = x^2 - x - 1