Composition of Linear Functions

Oct 13, 2025 by talynt.org 76 views

Introduction

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$ , the composition of $f$ and $g$ is denoted by $f \bullet g$ and is defined as $(f \bullet g)(x) = f(g(x))$ . In this article, we will explore the composition of two linear functions, $f(x) = -2x + 7$ and $g(x) = -6x + 3$ , and determine the resulting function $f \bullet g$ and its domain.

Composition of Linear Functions

To find the composition of $f$ and $g$ , we need to substitute $g(x)$ into $f(x)$ in place of $x$ . This means that we will replace every instance of $x$ in the function $f(x)$ with the expression $g(x) = -6x + 3$ . The resulting function will be a new function that combines the two original functions.

Finding $f \bullet g$

Let's substitute $g(x) = -6x + 3$ into $f(x) = -2x + 7$ :

$f(g(x)) = -2(-6x + 3) + 7$

Expanding the expression, we get:

$f(g(x)) = 12x - 6 + 7$

Combining like terms, we get:

$f(g(x)) = 12x + 1$

Therefore, the composition of $f$ and $g$ is $f \bullet g = 12x + 1$ .

Domain of $f \bullet g$

Since both $f(x)$ and $g(x)$ are linear functions, their domains are all real numbers. The domain of a function is the set of all possible input values for which the function is defined. In this case, both $f(x)$ and $g(x)$ are defined for all real numbers, so the domain of $f \bullet g$ is also all real numbers.

Conclusion

In conclusion, the composition of $f(x) = -2x + 7$ and $g(x) = -6x + 3$ is $f \bullet g = 12x + 1$ , and its domain is all real numbers. This demonstrates the concept of function composition and how it can be used to create new functions from existing ones.

Example 1: Composition of Functions

Suppose we have two functions $f(x) = 3x - 2$ and $g(x) = 2x + 1$ . Find the composition of $f$ and $g$ and state its domain.

Solution

To find the composition of $f$ and $g$ , we need to substitute $g(x)$ into $f(x)$ in place of $x$ . This means that we will replace every instance of $x$ in the function $f(x)$ with the expression $g(x) = 2x + 1$ . The resulting function will be a new function that combines the two original functions.

Let's substitute $g(x) = 2x + 1$ into $f(x) = 3x - 2$ :

$f(g(x)) = 3(2x + 1) - 2$

Expanding the expression, we get:

$f(g(x)) = 6x + 3 - 2$

Combining like terms, we get:

$f(g(x)) = 6x + 1$

Therefore, the composition of $f$ and $g$ is $f \bullet g = 6x + 1$ . Since both $f(x)$ and $g(x)$ are linear functions, their domains are all real numbers. Therefore, the domain of $f \bullet g$ is also all real numbers.

Example 2: Composition of Functions with Restrictions

Suppose we have two functions $f(x) = \frac{1}{x - 2}$ and $g(x) = 2x + 1$ . Find the composition of $f$ and $g$ and state its domain.

Solution

To find the composition of $f$ and $g$ , we need to substitute $g(x)$ into $f(x)$ in place of $x$ . This means that we will replace every instance of $x$ in the function $f(x)$ with the expression $g(x) = 2x + 1$ . The resulting function will be a new function that combines the two original functions.

Let's substitute $g(x) = 2x + 1$ into $f(x) = \frac{1}{x - 2}$ :

$f(g(x)) = \frac{1}{(2x + 1) - 2}$

Simplifying the expression, we get:

$f(g(x)) = \frac{1}{2x - 1}$

Therefore, the composition of $f$ and $g$ is $f \bullet g = \frac{1}{2x - 1}$ . Since $f(x)$ is not defined when $x = 2$ , we need to exclude $x = 2$ from the domain of $f \bullet g$ . Therefore, the domain of $f \bullet g$ is all real numbers except $x = 2$ .

Conclusion

In conclusion, the composition of functions is a powerful tool that allows us to create new functions from existing ones. By understanding how to find the composition of functions and their domains, we can solve a wide range of mathematical problems and applications.

Final Answer

The final answer is: $\boxed{12x + 1}$

Introduction

In our previous article, we explored the concept of composition of functions and how to find the composition of two linear functions. In this article, we will answer some frequently asked questions about composition of functions and provide additional examples to help you understand this concept better.

Q: What is the composition of functions?

A: The composition of functions is a way of combining two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$ , the composition of $f$ and $g$ is denoted by $f \bullet g$ and is defined as $(f \bullet g)(x) = f(g(x))$ .

Q: How do I find the composition of two functions?

A: To find the composition of two functions, you need to substitute the second function into the first function in place of $x$ . This means that you will replace every instance of $x$ in the first function with the expression of the second function.

Q: What is the domain of the composition of two functions?

A: The domain of the composition of two functions is the set of all possible input values for which the resulting function is defined. In general, the domain of the composition of two functions is the intersection of the domains of the two individual functions.

Q: Can I have a negative sign in the composition of two functions?

A: Yes, you can have a negative sign in the composition of two functions. For example, if we have two functions $f(x) = -2x + 7$ and $g(x) = -6x + 3$ , the composition of $f$ and $g$ is $f \bullet g = 12x + 1$ .

Q: Can I have a fraction in the composition of two functions?

A: Yes, you can have a fraction in the composition of two functions. For example, if we have two functions $f(x) = \frac{1}{x - 2}$ and $g(x) = 2x + 1$ , the composition of $f$ and $g$ is $f \bullet g = \frac{1}{2x - 1}$ .

Q: Can I have a square root in the composition of two functions?

A: Yes, you can have a square root in the composition of two functions. For example, if we have two functions $f(x) = \sqrt{x - 2}$ and $g(x) = 2x + 1$ , the composition of $f$ and $g$ is $f \bullet g = \sqrt{2x - 1}$ .

Q: Can I have a logarithm in the composition of two functions?

A: Yes, you can have a logarithm in the composition of two functions. For example, if we have two functions $f(x) = \log(x - 2)$ and $g(x) = 2x + 1$ , the composition of $f$ and $g$ is $f \bullet g = \log(2x - 1)$ .

Q: Can I have a trigonometric function in the composition of two functions?

A: Yes, you can have a trigonometric function in the composition of two functions. For example, if we have two functions $f(x) = \sin(x - 2)$ and $g(x) = 2x + 1$ , the composition of $f$ and $g$ is $f \bullet g = \sin(2x - 1)$ .

Q: Can I have a polynomial function in the composition of two functions?

A: Yes, you can have a polynomial function in the composition of two functions. For example, if we have two functions $f(x) = x^2 + 2x + 1$ and $g(x) = 2x + 1$ , the composition of $f$ and $g$ is $f \bullet g = (2x + 1)^2 + 2(2x + 1) + 1$ .

Conclusion

Final Answer

The final answer is: $\boxed{12x + 1}$