The Function \[$ F \$\] Is Shown Below:$\[ F(x) = \frac{1}{x-1} + \frac{1}{x-4} \\]Work Out The Value Of \[$ F(9) \$\]. Give Your Answer As A Fraction In Its Simplest Form.
Introduction
In mathematics, functions are a fundamental concept that help us describe the relationship between variables. A function is a rule that assigns to each input, or value, a unique output. In this article, we will explore the function f(x) and evaluate its value at x = 9.
The Function f(x)
The function f(x) is given by the equation:
This function consists of two terms, each of which is a fraction. The first term is 1/(x-1), and the second term is 1/(x-4). To evaluate the function at a specific value of x, we need to substitute that value into the equation and simplify.
Evaluating f(9)
To evaluate f(9), we need to substitute x = 9 into the equation for f(x). This gives us:
Adding Fractions with Different Denominators
When we add fractions with different denominators, we need to find a common denominator. In this case, the least common multiple (LCM) of 8 and 5 is 40. We can rewrite each fraction with a denominator of 40:
Simplifying the Expression
Now that we have a common denominator, we can add the fractions:
Conclusion
In this article, we evaluated the function f(x) at x = 9. We substituted x = 9 into the equation for f(x), simplified the expression, and found that f(9) = 13/40.
Importance of Evaluating Functions
Evaluating functions is an essential skill in mathematics, as it allows us to understand the behavior of functions and make predictions about their values. In real-world applications, functions are used to model a wide range of phenomena, from population growth to electrical circuits. By evaluating functions, we can gain insights into the underlying mechanisms and make informed decisions.
Real-World Applications of Functions
Functions have numerous real-world applications, including:
- Physics and Engineering: Functions are used to describe the motion of objects, the flow of fluids, and the behavior of electrical circuits.
- Economics: Functions are used to model economic systems, including supply and demand curves, and to make predictions about economic trends.
- Biology: Functions are used to model population growth, the spread of diseases, and the behavior of ecosystems.
Tips for Evaluating Functions
When evaluating functions, it's essential to follow these tips:
- Read the problem carefully: Make sure you understand what the problem is asking and what function is being evaluated.
- Substitute the value: Substitute the given value into the equation for the function.
- Simplify the expression: Simplify the expression by combining like terms and canceling out any common factors.
- Check your answer: Check your answer to make sure it's reasonable and makes sense in the context of the problem.
Common Mistakes to Avoid
When evaluating functions, it's essential to avoid these common mistakes:
- Not reading the problem carefully: Make sure you understand what the problem is asking and what function is being evaluated.
- Not substituting the value: Substitute the given value into the equation for the function.
- Not simplifying the expression: Simplify the expression by combining like terms and canceling out any common factors.
- Not checking your answer: Check your answer to make sure it's reasonable and makes sense in the context of the problem.
Conclusion
In conclusion, evaluating functions is an essential skill in mathematics that has numerous real-world applications. By following the tips and avoiding common mistakes, you can become proficient in evaluating functions and make informed decisions in a wide range of fields.
Introduction
Evaluating functions is a fundamental concept in mathematics that has numerous real-world applications. In our previous article, we explored the function f(x) and evaluated its value at x = 9. In this article, we will answer some common questions about evaluating functions and provide additional tips and resources for mastering this skill.
Q&A: Evaluating Functions
Q: What is the difference between evaluating a function and solving an equation?
A: Evaluating a function involves substituting a specific value into the equation for the function and simplifying the expression. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true.
Q: How do I know when to evaluate a function?
A: You should evaluate a function when you are given a specific value to substitute into the equation for the function. This is often indicated by a question or problem that asks you to find the value of the function at a specific point.
Q: What are some common mistakes to avoid when evaluating functions?
A: Some common mistakes to avoid when evaluating functions include:
- Not reading the problem carefully
- Not substituting the value into the equation for the function
- Not simplifying the expression
- Not checking your answer to make sure it's reasonable and makes sense in the context of the problem
Q: How do I simplify an expression when evaluating a function?
A: To simplify an expression when evaluating a function, you should combine like terms and cancel out any common factors. You can also use algebraic properties, such as the distributive property, to simplify the expression.
Q: What are some real-world applications of evaluating functions?
A: Evaluating functions has numerous real-world applications, including:
- Physics and Engineering: Functions are used to describe the motion of objects, the flow of fluids, and the behavior of electrical circuits.
- Economics: Functions are used to model economic systems, including supply and demand curves, and to make predictions about economic trends.
- Biology: Functions are used to model population growth, the spread of diseases, and the behavior of ecosystems.
Tips for Evaluating Functions
Tip 1: Read the problem carefully
Make sure you understand what the problem is asking and what function is being evaluated.
Tip 2: Substitute the value
Substitute the given value into the equation for the function.
Tip 3: Simplify the expression
Simplify the expression by combining like terms and canceling out any common factors.
Tip 4: Check your answer
Check your answer to make sure it's reasonable and makes sense in the context of the problem.
Additional Resources
- Khan Academy: Evaluating Functions
- Mathway: Evaluating Functions
- Wolfram Alpha: Evaluating Functions
Conclusion
Evaluating functions is a fundamental concept in mathematics that has numerous real-world applications. By following the tips and avoiding common mistakes, you can become proficient in evaluating functions and make informed decisions in a wide range of fields. Remember to read the problem carefully, substitute the value, simplify the expression, and check your answer to ensure that you are evaluating functions correctly.
Frequently Asked Questions
Q: What is the difference between a function and an equation?
A: A function is a rule that assigns to each input, or value, a unique output. An equation is a statement that two expressions are equal.
Q: How do I know when to use a function versus an equation?
A: You should use a function when you are given a specific value to substitute into the equation for the function. You should use an equation when you are trying to find the value of the variable that makes the equation true.
Q: What are some common types of functions?
A: Some common types of functions include:
- Linear functions
- Quadratic functions
- Polynomial functions
- Rational functions
Q: How do I graph a function?
A: To graph a function, you should first identify the type of function and its characteristics. Then, you can use a graphing calculator or software to visualize the function.
Conclusion
Evaluating functions is a fundamental concept in mathematics that has numerous real-world applications. By following the tips and avoiding common mistakes, you can become proficient in evaluating functions and make informed decisions in a wide range of fields. Remember to read the problem carefully, substitute the value, simplify the expression, and check your answer to ensure that you are evaluating functions correctly.