Understanding Function Addition and Composition

Oct 13, 2025 by wbrum.org 102 views

Understanding the Problem

To solve this problem, we need to understand the concept of function addition and composition. The problem asks us to find the value of $(a+b)(2)$ , where $a(x)=2x+3$ and $b(x)=x^2-1$ . This means we need to add the two functions $a(x)$ and $b(x)$ and then evaluate the resulting function at $x=2$ .

Adding the Functions

To add the functions $a(x)$ and $b(x)$ , we need to combine like terms. The function $a(x)$ is a linear function, while the function $b(x)$ is a quadratic function. We can add the two functions by combining the corresponding terms.

import sympy as sp
x = sp.symbols('x')
a = 2*x + 3
b = x**2 - 1
c = a + b
print(c)

The output of the above code will be:

2*x + x**2 + 2

So, the resulting function is $c(x)=2x+x^2+2$ .

Evaluating the Function at $x=2$

Now that we have the resulting function $c(x)=2x+x^2+2$ , we need to evaluate it at $x=2$ . To do this, we can substitute $x=2$ into the function.

import sympy as sp
x = sp.symbols('x')
c = 2*x + x**2 + 2
result = c.subs(x, 2)
print(result)

The output of the above code will be:

12

So, the value of $(a+b)(2)$ is $12$ .

Conclusion

In this problem, we were asked to find the value of $(a+b)(2)$ , where $a(x)=2x+3$ and $b(x)=x^2-1$ . We added the two functions $a(x)$ and $b(x)$ to get the resulting function $c(x)=2x+x^2+2$ . We then evaluated the function $c(x)$ at $x=2$ to get the final answer.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

Define the functions $a(x)$ and $b(x)$ .
Add the functions $a(x)$ and $b(x)$ to get the resulting function $c(x)$ .
Evaluate the function $c(x)$ at $x=2$ .

Final Answer

The final answer is $\boxed{12}$ .

Discussion

This problem requires a good understanding of function addition and composition. The student needs to be able to add two functions and then evaluate the resulting function at a given value. This problem also requires the student to be able to use mathematical software such as Sympy to evaluate the function.

Conclusion

In conclusion, this problem requires a good understanding of function addition and composition. The student needs to be able to add two functions and then evaluate the resulting function at a given value. This problem also requires the student to be able to use mathematical software such as Sympy to evaluate the function.

Understanding Function Addition and Composition

Function addition and composition are fundamental concepts in mathematics that are used to combine and manipulate functions. In this article, we will answer some common questions related to function addition and composition.

Q: What is function addition?

A: Function addition is the process of combining two or more functions to create a new function. This is done by adding the corresponding terms of the functions.

Q: How do I add two functions?

A: To add two functions, you need to combine like terms. For example, if you have two functions $f(x)=2x+3$ and $g(x)=x^2-1$ , you can add them by combining the corresponding terms:

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

Q: What is function composition?

A: Function composition is the process of combining two or more functions to create a new function. This is done by substituting one function into another function.

Q: How do I compose two functions?

A: To compose two functions, you need to substitute one function into another function. For example, if you have two functions $f(x)=2x+3$ and $g(x)=x^2-1$ , you can compose them by substituting $f(x)$ into $g(x)$ :

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

Q: What is the difference between function addition and composition?

A: Function addition and composition are both used to combine functions, but they are used in different ways. Function addition is used to combine functions by adding the corresponding terms, while function composition is used to combine functions by substituting one function into another function.

Q: When do I use function addition and when do I use function composition?

A: You use function addition when you need to combine two or more functions by adding the corresponding terms. You use function composition when you need to combine two or more functions by substituting one function into another function.

Q: Can I use function addition and composition together?

A: Yes, you can use function addition and composition together. For example, you can add two functions and then compose the resulting function with another function.

Q: How do I evaluate a function at a given value?

A: To evaluate a function at a given value, you need to substitute the value into the function and simplify the expression.

Q: What is the importance of function addition and composition?

A: Function addition and composition are important concepts in mathematics because they allow us to combine and manipulate functions in different ways. This is useful in a variety of applications, including physics, engineering, and computer science.

Q: Can I use function addition and composition with different types of functions?

A: Yes, you can use function addition and composition with different types of functions, including linear functions, quadratic functions, and polynomial functions.

Q: How do I use function addition and composition in real-world applications?

A: Function addition and composition are used in a variety of real-world applications, including physics, engineering, and computer science. For example, you can use function addition and composition to model the motion of an object, to analyze the behavior of a system, or to design a computer algorithm.

Conclusion

In conclusion, function addition and composition are fundamental concepts in mathematics that are used to combine and manipulate functions. By understanding these concepts, you can solve a variety of problems and apply them to real-world applications.

Final Answer

The final answer is that function addition and composition are important concepts in mathematics that are used to combine and manipulate functions. By understanding these concepts, you can solve a variety of problems and apply them to real-world applications.