Recalling Function Addition

Oct 13, 2025 by miodeya.com 111 views

If $f(x)=\frac{x}{2}-3$ and $g(x)=4x^2+x-4$ , find $(f+g)(x)$

Understanding the Problem

To find the sum of two functions, we need to add their corresponding terms. In this case, we are given two functions, $f(x)=\frac{x}{2}-3$ and $g(x)=4x^2+x-4$ . We are asked to find the sum of these two functions, denoted as $(f+g)(x)$ .

Recalling Function Addition

When adding two functions, we add their corresponding terms. This means that we add the terms with the same variable and coefficient. In this case, we have:

$f(x)=\frac{x}{2}-3$ $g(x)=4x^2+x-4$

To find the sum of these two functions, we add their corresponding terms:

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

Adding the Terms

Now, let's add the terms of the two functions:

$(f+g)(x)=\frac{x}{2}-3+4x^2+x-4$

To add these terms, we need to combine like terms. We can start by combining the constant terms:

$-3-4=-7$

Next, we can combine the terms with the variable $x$ :

$\frac{x}{2}+x$

To combine these terms, we need to find a common denominator. In this case, the common denominator is 2. So, we can rewrite the terms as:

$\frac{x}{2}+\frac{2x}{2}$

Now, we can add the terms:

$\frac{x}{2}+\frac{2x}{2}=\frac{3x}{2}$

Simplifying the Expression

Now that we have added the terms, we can simplify the expression:

$(f+g)(x)=4x^2+\frac{3x}{2}-7$

Evaluating the Answer Choices

Now that we have found the sum of the two functions, we can evaluate the answer choices:

A. $4x^2+\frac{1}{2}x-1$ B. $4x^2+\frac{3}{2}x-1$ C. $6x^2-7$ D. $4x^2+\frac{3}{2}x-7$

Comparing our answer with the answer choices, we can see that the correct answer is:

D. $4x^2+\frac{3}{2}x-7$

Conclusion

In this problem, we were asked to find the sum of two functions, $f(x)=\frac{x}{2}-3$ and $g(x)=4x^2+x-4$ . We added the corresponding terms of the two functions and simplified the expression to find the sum. The correct answer is:

D. $4x^2+\frac{3}{2}x-7$

Key Takeaways

When adding two functions, we add their corresponding terms.
To add terms with the same variable and coefficient, we combine like terms.
To combine like terms, we need to find a common denominator.
When simplifying an expression, we can combine like terms and eliminate any common factors.

Practice Problems

Find the sum of the functions $f(x)=2x^2-3x+1$ and $g(x)=x^2+2x-4$ .
Find the sum of the functions $f(x)=\frac{x}{3}-2$ and $g(x)=2x^2+x-1$ .

Solutions

$(f+g)(x)=3x^2-5x-3$
$(f+g)(x)=2x^2+\frac{1}{3}x-3$
Q&A: Function Addition and Simplification

Understanding Function Addition

Q: What is function addition? A: Function addition is the process of adding two or more functions together. When adding functions, we add their corresponding terms.

Q: How do I add functions? A: To add functions, we add their corresponding terms. This means that we add the terms with the same variable and coefficient.

Simplifying Expressions

Q: How do I simplify an expression after adding functions? A: To simplify an expression, we can combine like terms and eliminate any common factors. We can also rewrite the expression in a more compact form.

Q: What are like terms? A: Like terms are terms that have the same variable and coefficient. For example, $2x$ and $3x$ are like terms because they both have the variable $x$ and the coefficient $2$ and $3$ respectively.

Common Mistakes

Q: What are some common mistakes to avoid when adding functions? A: Some common mistakes to avoid when adding functions include:

Not combining like terms
Not eliminating common factors
Not rewriting the expression in a more compact form

Tips and Tricks

Q: What are some tips and tricks for adding functions? A: Some tips and tricks for adding functions include:

Use a common denominator when adding fractions
Combine like terms before simplifying the expression
Eliminate common factors to simplify the expression

Practice Problems

Find the sum of the functions $f(x)=2x^2-3x+1$ and $g(x)=x^2+2x-4$ .
Find the sum of the functions $f(x)=\frac{x}{3}-2$ and $g(x)=2x^2+x-1$ .

Solutions

$(f+g)(x)=3x^2-5x-3$
$(f+g)(x)=2x^2+\frac{1}{3}x-3$

Real-World Applications

Q: How are functions used in real-world applications? A: Functions are used in a wide range of real-world applications, including:

Physics: Functions are used to describe the motion of objects and the behavior of physical systems.
Engineering: Functions are used to design and optimize systems, such as bridges and buildings.
Economics: Functions are used to model economic systems and make predictions about future trends.

Conclusion

In this article, we have discussed the basics of function addition and simplification. We have also covered some common mistakes to avoid and some tips and tricks for adding functions. We have also provided some practice problems and solutions to help you practice your skills.