Use The Following Function Rule To Find $f(-6)$:$f(x) = 9 - X$$ F ( − 6 ) = F(-6) = F ( − 6 ) = [/tex]

Understanding the Function Rule

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function rule is a mathematical expression that describes how to calculate the output for a given input. In this article, we will use the function rule $f(x) = 9 - x$ to find the value of $f(-6)$.

The Function Rule: $f(x) = 9 - x$

The function rule $f(x) = 9 - x$ is a linear function, which means that it can be represented by a straight line. The function has a constant term of 9 and a coefficient of -1 for the variable x. This means that for every unit increase in x, the value of the function decreases by 1 unit.

Substituting x = -6 into the Function Rule

To find the value of $f(-6)$, we need to substitute x = -6 into the function rule. This means that we will replace x with -6 in the expression $f(x) = 9 - x$.

Calculating the Value of $f(-6)$

Now that we have substituted x = -6 into the function rule, we can calculate the value of $f(-6)$. We will do this by evaluating the expression $f(-6) = 9 - (-6)$.

Simplifying the Expression

To simplify the expression $f(-6) = 9 - (-6)$, we need to remember that subtracting a negative number is the same as adding a positive number. Therefore, we can rewrite the expression as $f(-6) = 9 + 6$.

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it to find the value of $f(-6)$. We will do this by adding 9 and 6.

The Final Answer

The final answer is: $\boxed{15}$

Conclusion

In this article, we used the function rule $f(x) = 9 - x$ to find the value of $f(-6)$. We substituted x = -6 into the function rule, simplified the expression, and evaluated it to find the final answer. This demonstrates the importance of following the order of operations and using the correct mathematical properties to solve problems.

Common Mistakes to Avoid

When working with functions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations: When evaluating expressions, it's essential to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
• Not using the correct mathematical properties: Remember that subtracting a negative number is the same as adding a positive number.
• Not checking the domain and range: Make sure that the input value is within the domain of the function and that the output value is within the range of the function.

Real-World Applications

Functions are used in many real-world applications, including:

• Physics and Engineering: Functions are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
• Economics: Functions are used to model the behavior of economic systems, including supply and demand curves and cost-benefit analyses.
• Computer Science: Functions are used to write algorithms, implement data structures, and create software applications.

Tips and Tricks

Here are some tips and tricks to help you work with functions:

• Use a calculator: When working with functions, it's often helpful to use a calculator to evaluate expressions and check your work.
• Graph functions: Graphing functions can help you visualize the behavior of the function and identify key features such as the x-intercept, y-intercept, and vertex.
• Check your work: Always check your work to ensure that you have followed the correct mathematical properties and have used the correct order of operations.

Conclusion

In conclusion, functions are an essential part of mathematics and are used in many real-world applications. By following the function rule, substituting values, simplifying expressions, and evaluating the final answer, you can find the value of a function. Remember to avoid common mistakes, use a calculator, graph functions, and check your work to ensure that you are working with functions correctly.

Understanding Functions: A Q&A Guide

Functions are a fundamental concept in mathematics, and they have numerous real-world applications. However, they can be challenging to understand and work with, especially for beginners. In this article, we will provide a comprehensive Q&A guide to help you master functions.

Q: What is a function?

A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing how to calculate the output for a given input.

Q: What is the difference between a function and an equation?

A: An equation is a statement that two expressions are equal, while a function is a relation between a set of inputs and a set of possible outputs. In other words, an equation is a statement, while a function is a rule.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible y-values.

Q: How do I find the value of a function?

A: To find the value of a function, you need to substitute the input value into the function rule and evaluate the expression.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when evaluating an expression. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I simplify an expression?

A: To simplify an expression, you need to combine like terms and eliminate any unnecessary parentheses.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a function that can be represented by a straight line, while a quadratic function is a function that can be represented by a parabola.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on a coordinate plane and draw a smooth curve through the points.

Q: What is the x-intercept of a function?

A: The x-intercept of a function is the point where the function crosses the x-axis. In other words, it is the point where the output value is zero.

Q: What is the y-intercept of a function?

A: The y-intercept of a function is the point where the function crosses the y-axis. In other words, it is the point where the input value is zero.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to use the formula x = -b/2a, where a and b are the coefficients of the quadratic function.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs and a set of possible outputs, while a relation is a set of ordered pairs.

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you need to check if each input value corresponds to only one output value.

Conclusion

In conclusion, functions are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding the basics of functions, you can master this concept and apply it to various fields. Remember to practice, practice, practice, and don't be afraid to ask questions.

Common Mistakes to Avoid

When working with functions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations: When evaluating expressions, it's essential to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
• Not using the correct mathematical properties: Remember that subtracting a negative number is the same as adding a positive number.
• Not checking the domain and range: Make sure that the input value is within the domain of the function and that the output value is within the range of the function.

Real-World Applications

Functions are used in many real-world applications, including:

• Physics and Engineering: Functions are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
• Economics: Functions are used to model the behavior of economic systems, including supply and demand curves and cost-benefit analyses.
• Computer Science: Functions are used to write algorithms, implement data structures, and create software applications.

Tips and Tricks

Here are some tips and tricks to help you work with functions:

• Use a calculator: When working with functions, it's often helpful to use a calculator to evaluate expressions and check your work.
• Graph functions: Graphing functions can help you visualize the behavior of the function and identify key features such as the x-intercept, y-intercept, and vertex.
• Check your work: Always check your work to ensure that you have followed the correct mathematical properties and have used the correct order of operations.