Given The Functions:$\[ \begin{align} f(x) &= X^2 + 9 \\ g(x) &= X - 9 \end{align} \\]Find \[$ F(x) - G(x) \$\].

Feb 27, 2025 by ADMIN 115 views

**Finding the Difference Between Two Functions**

Introduction

In mathematics, functions are used to describe relationships between variables. Given two functions, we can find the difference between them by subtracting one function from the other. In this article, we will explore how to find the difference between two given functions, $f(x)$ and $g(x)$ .

The Functions

The two functions given are:

f(x) = x^2 + 9
g(x) = x - 9

Finding the Difference

To find the difference between the two functions, we need to subtract $g(x)$ from $f(x)$ . This can be done by subtracting the corresponding terms of the two functions.

Step 1: Subtract the x^2 Terms

The first step is to subtract the $x^2$ terms of the two functions. Since $f(x)$ has an $x^2$ term and $g(x)$ does not, we can simply write:

f(x) - g(x) = (x^2 + 9) - (x - 9)

Step 2: Distribute the Negative Sign

Next, we need to distribute the negative sign to the terms inside the parentheses. This will change the sign of each term.

f(x) - g(x) = x^2 + 9 - x + 9

Step 3: Combine Like Terms

Now, we can combine the like terms. The two constant terms, 9 and 9, can be combined into a single term.

f(x) - g(x) = x^2 - x + 18

The Final Answer

Therefore, the difference between the two functions, $f(x)$ and $g(x)$ , is:

f(x) - g(x) = x^2 - x + 18

Conclusion

In this article, we found the difference between two given functions, $f(x)$ and $g(x)$ . We used the properties of functions to subtract one function from the other and obtained the final answer. This demonstrates the importance of understanding the properties of functions in mathematics.

Example Use Cases

The difference between two functions can be used in a variety of applications, such as:

Optimization: Finding the difference between two functions can help us optimize a system or process.
Modeling: The difference between two functions can be used to model real-world phenomena, such as population growth or chemical reactions.
Data Analysis: The difference between two functions can be used to analyze data and identify trends or patterns.

Tips and Tricks

When finding the difference between two functions, remember to:

Distribute the negative sign: When subtracting one function from another, make sure to distribute the negative sign to the terms inside the parentheses.
Combine like terms: Combine the like terms to simplify the expression.
Check your work: Double-check your work to ensure that you have obtained the correct answer.

Common Mistakes

When finding the difference between two functions, some common mistakes to avoid include:

Forgetting to distribute the negative sign: Make sure to distribute the negative sign to the terms inside the parentheses.
Not combining like terms: Combine the like terms to simplify the expression.
Not checking your work: Double-check your work to ensure that you have obtained the correct answer.

Conclusion

Introduction

In our previous article, we explored how to find the difference between two functions, $f(x)$ and $g(x)$ . In this article, we will answer some common questions related to finding the difference between two functions.

Q: What is the difference between subtracting two functions and finding the difference between two functions?

A: Subtracting two functions means finding the value of one function minus the value of another function at a given point. Finding the difference between two functions, on the other hand, means finding the expression that represents the difference between the two functions.

Q: How do I find the difference between two functions with different variables?

A: To find the difference between two functions with different variables, you need to substitute the variables of one function into the other function. For example, if you have two functions, $f(x)$ and $g(y)$ , you can find the difference between them by substituting $y$ for $x$ in $f(x)$ .

Q: Can I find the difference between two functions with different domains?

A: Yes, you can find the difference between two functions with different domains. However, you need to make sure that the domains of the two functions overlap. If the domains do not overlap, you cannot find the difference between the two functions.

Q: How do I find the difference between two functions with different ranges?

A: To find the difference between two functions with different ranges, you need to make sure that the ranges of the two functions overlap. If the ranges do not overlap, you cannot find the difference between the two functions.

Q: Can I find the difference between two functions that are not defined at the same point?

A: Yes, you can find the difference between two functions that are not defined at the same point. However, you need to make sure that the functions are defined at all points except at the point where they are not defined.

Q: How do I find the difference between two functions that are defined on different intervals?

A: To find the difference between two functions that are defined on different intervals, you need to make sure that the intervals overlap. If the intervals do not overlap, you cannot find the difference between the two functions.

Q: Can I find the difference between two functions that are not continuous?

A: Yes, you can find the difference between two functions that are not continuous. However, you need to make sure that the functions are defined at all points except at the points where they are not continuous.

Q: How do I find the difference between two functions that are defined on different coordinate systems?

A: To find the difference between two functions that are defined on different coordinate systems, you need to make sure that the coordinate systems are equivalent. If the coordinate systems are not equivalent, you cannot find the difference between the two functions.

Conclusion

In conclusion, finding the difference between two functions is an important concept in mathematics. By understanding the properties of functions and following the steps outlined in this article, we can find the difference between two functions and apply it to a variety of applications.

Tips and Tricks

When finding the difference between two functions, remember to:

Check the domains and ranges: Make sure that the domains and ranges of the two functions overlap.
Substitute variables: Substitute the variables of one function into the other function.
Make sure the functions are defined: Make sure that the functions are defined at all points except at the points where they are not defined.
Use equivalent coordinate systems: Use equivalent coordinate systems to find the difference between two functions that are defined on different coordinate systems.

Common Mistakes

When finding the difference between two functions, some common mistakes to avoid include:

Not checking the domains and ranges: Make sure that the domains and ranges of the two functions overlap.
Not substituting variables: Substitute the variables of one function into the other function.
Not making sure the functions are defined: Make sure that the functions are defined at all points except at the points where they are not defined.
Not using equivalent coordinate systems: Use equivalent coordinate systems to find the difference between two functions that are defined on different coordinate systems.