When $f(x) = 9 - 3x$ And $g(x) = 5x - 7$, Find \$f(x) + G(x)$[/tex\].
In the realm of mathematics, functions play a vital role in describing relationships between variables. When we encounter two functions, f(x) and g(x), we often need to combine them to form a new function. In this article, we will explore the process of combining two functions, f(x) = 9 - 3x and g(x) = 5x - 7, to find their sum, f(x) + g(x).
Understanding the Functions
Before we dive into combining the functions, let's take a closer look at each function individually.
f(x) = 9 - 3x
The function f(x) = 9 - 3x is a linear function that takes a single input, x, and produces an output. The function has a constant term, 9, and a coefficient, -3, which is multiplied by the input x. This function represents a straight line with a negative slope.
g(x) = 5x - 7
The function g(x) = 5x - 7 is also a linear function that takes a single input, x, and produces an output. The function has a coefficient, 5, which is multiplied by the input x, and a constant term, -7. This function represents a straight line with a positive slope.
Combining the Functions
Now that we have a good understanding of each function, let's combine them to find their sum, f(x) + g(x).
To combine the functions, we need to add the corresponding terms. We can start by adding the constant terms:
f(x) + g(x) = (9 - 3x) + (5x - 7)
Next, we can combine the like terms:
f(x) + g(x) = 9 - 3x + 5x - 7
Now, we can simplify the expression by combining the x terms:
f(x) + g(x) = 9 + 2x - 7
Finally, we can simplify the expression by combining the constant terms:
f(x) + g(x) = 2x + 2
Simplifying the Expression
The expression f(x) + g(x) = 2x + 2 is a simplified form of the combined function. We can rewrite this expression in a more concise form:
f(x) + g(x) = 2(x + 1)
This form makes it clear that the combined function is a linear function with a coefficient of 2 and a constant term of 2.
Conclusion
In this article, we combined two functions, f(x) = 9 - 3x and g(x) = 5x - 7, to find their sum, f(x) + g(x). We started by adding the corresponding terms and then simplified the expression by combining like terms. The resulting expression, f(x) + g(x) = 2x + 2, is a simplified form of the combined function. We also rewrote this expression in a more concise form, f(x) + g(x) = 2(x + 1), which makes it clear that the combined function is a linear function with a coefficient of 2 and a constant term of 2.
Real-World Applications
The process of combining functions is a fundamental concept in mathematics that has numerous real-world applications. In physics, for example, we often need to combine functions to describe the motion of objects. In economics, we may need to combine functions to model the behavior of markets. In computer science, we may need to combine functions to implement algorithms.
Tips and Tricks
When combining functions, it's essential to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
By following this order of operations, you can ensure that your combined function is accurate and reliable.
Common Mistakes
When combining functions, it's easy to make mistakes. Here are a few common mistakes to watch out for:
- Forgetting to combine like terms: Make sure to combine any like terms, such as x terms or constant terms.
- Not following the order of operations: Make sure to follow the order of operations (PEMDAS) to ensure that your combined function is accurate and reliable.
- Not simplifying the expression: Make sure to simplify the expression by combining like terms and eliminating any unnecessary terms.
By avoiding these common mistakes, you can ensure that your combined function is accurate and reliable.
Conclusion
In our previous article, we explored the process of combining two functions, f(x) = 9 - 3x and g(x) = 5x - 7, to find their sum, f(x) + g(x). In this article, we will answer some frequently asked questions about combining functions.
Q: What is the purpose of combining functions?
A: The purpose of combining functions is to create a new function that represents the sum or difference of two or more individual functions. This can be useful in a variety of mathematical and real-world applications, such as modeling the behavior of physical systems or analyzing economic data.
Q: How do I combine two functions?
A: To combine two functions, you need to add or subtract the corresponding terms. For example, if you want to find the sum of f(x) = 9 - 3x and g(x) = 5x - 7, you would add the constant terms and combine the like terms.
Q: What is the order of operations when combining functions?
A: When combining functions, you need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I simplify the expression after combining functions?
A: To simplify the expression after combining functions, you need to combine like terms and eliminate any unnecessary terms. For example, if you have the expression f(x) + g(x) = 9 - 3x + 5x - 7, you can simplify it by combining the x terms and the constant terms.
Q: What are some common mistakes to watch out for when combining functions?
A: Some common mistakes to watch out for when combining functions include:
- Forgetting to combine like terms
- Not following the order of operations
- Not simplifying the expression
- Not checking for any errors or inconsistencies in the expression
Q: Can I combine more than two functions?
A: Yes, you can combine more than two functions. The process is similar to combining two functions, but you need to add or subtract the corresponding terms from each function.
Q: How do I know if the combined function is accurate and reliable?
A: To ensure that the combined function is accurate and reliable, you need to follow the order of operations and simplify the expression. You should also check for any errors or inconsistencies in the expression.
Q: Can I use combining functions in real-world applications?
A: Yes, combining functions has numerous real-world applications, such as modeling the behavior of physical systems, analyzing economic data, and implementing algorithms.
Q: What are some examples of real-world applications of combining functions?
A: Some examples of real-world applications of combining functions include:
- Modeling the motion of objects in physics
- Analyzing economic data in economics
- Implementing algorithms in computer science
- Predicting the behavior of complex systems in engineering
Conclusion
In conclusion, combining functions is a fundamental concept in mathematics that has numerous real-world applications. By following the order of operations and simplifying the expression, you can ensure that your combined function is accurate and reliable. Remember to watch out for common mistakes, such as forgetting to combine like terms or not following the order of operations. With practice and patience, you can become proficient in combining functions and apply this skill to a wide range of mathematical and real-world problems.