Evaluating the Inner Function

Oct 13, 2025 by windailys.com 21 views

If $f(x) = 7 + 4x$ and $g(x) = \frac{1}{2x}$ , what is the value of $\left(\frac{f}{g}\right)(5)$ ?

Understanding the Problem

The problem involves two functions, $f(x)$ and $g(x)$ , and asks us to find the value of the composite function $\left(\frac{f}{g}\right)(5)$ . To solve this problem, we need to understand the concept of composite functions and how to evaluate them.

Composite Functions

A composite function is a function that is defined in terms of another function. In this case, we have two functions, $f(x)$ and $g(x)$ , and we want to find the value of the composite function $\left(\frac{f}{g}\right)(5)$ . To evaluate this composite function, we need to follow the order of operations and first evaluate the inner function, $g(5)$ .

Evaluating the Inner Function

To evaluate the inner function, $g(5)$ , we need to substitute $x = 5$ into the function $g(x) = \frac{1}{2x}$ . This gives us:

g(5) = \frac{1}{2(5)} = \frac{1}{10}

Evaluating the Composite Function

Now that we have evaluated the inner function, $g(5)$ , we can evaluate the composite function $\left(\frac{f}{g}\right)(5)$ . To do this, we need to substitute $g(5) = \frac{1}{10}$ into the function $f(x) = 7 + 4x$ . This gives us:

\left(\frac{f}{g}\right)(5) = \frac{f(5)}{g(5)} = \frac{7 + 4(5)}{\frac{1}{10}} = \frac{7 + 20}{\frac{1}{10}} = \frac{27}{\frac{1}{10}} = 27 \times 10 = 270

Conclusion

Therefore, the value of $\left(\frac{f}{g}\right)(5)$ is $\boxed{270}$ .

Step-by-Step Solution

Here is the step-by-step solution to the problem:

Evaluate the inner function, $g(5)$ , by substituting $x = 5$ into the function $g(x) = \frac{1}{2x}$ .
Evaluate the composite function $\left(\frac{f}{g}\right)(5)$ by substituting $g(5) = \frac{1}{10}$ into the function $f(x) = 7 + 4x$ .
Simplify the expression to find the final answer.

Key Concepts

Composite functions
Evaluating inner functions
Evaluating composite functions
Simplifying expressions

Real-World Applications

This problem can be applied to real-world situations where we need to evaluate composite functions, such as in physics, engineering, and economics.
Understanding composite functions is crucial in solving problems that involve multiple functions and variables.

Common Mistakes

Failing to evaluate the inner function before evaluating the composite function.
Not simplifying the expression correctly.
Not following the order of operations.

Tips and Tricks

Make sure to evaluate the inner function before evaluating the composite function.
Simplify the expression as much as possible to avoid errors.
Follow the order of operations to ensure accuracy.

Practice Problems

Evaluate the composite function $\left(\frac{f}{g}\right)(x)$ , where $f(x) = 2x + 1$ and $g(x) = \frac{1}{x}$ .
Evaluate the composite function $\left(\frac{f}{g}\right)(x)$ , where $f(x) = x^2 + 1$ and $g(x) = \frac{1}{x^2}$ .

Conclusion

In conclusion, the value of $\left(\frac{f}{g}\right)(5)$ is $\boxed{270}$ . This problem requires understanding composite functions, evaluating inner functions, and simplifying expressions. By following the step-by-step solution and key concepts, we can solve this problem and apply it to real-world situations.
Q&A: Composite Functions

Understanding Composite Functions

Composite functions are a fundamental concept in mathematics that can be used to solve a wide range of problems. In this article, we will answer some of the most frequently asked questions about composite functions.

Q: What is a composite function?

A: A composite function is a function that is defined in terms of another function. It is a function that takes another function as its input and produces a new function as its output.

Q: How do I evaluate a composite function?

A: To evaluate a composite function, you need to follow the order of operations and first evaluate the inner function. Then, substitute the result into the outer function and simplify the expression.

Q: What is the difference between a composite function and a function of a function?

A: A composite function is a function that takes another function as its input and produces a new function as its output. A function of a function, on the other hand, is a function that takes a value as its input and produces a value as its output.

Q: Can I use composite functions to solve real-world problems?

A: Yes, composite functions can be used to solve a wide range of real-world problems. For example, in physics, composite functions can be used to model the motion of objects. In economics, composite functions can be used to model the behavior of markets.

Q: How do I know when to use a composite function?

A: You should use a composite function when you need to evaluate a function that takes another function as its input. This can be the case in a wide range of situations, including physics, engineering, and economics.

Q: What are some common mistakes to avoid when working with composite functions?

A: Some common mistakes to avoid when working with composite functions include:

Failing to evaluate the inner function before evaluating the composite function
Not simplifying the expression correctly
Not following the order of operations

Q: How can I practice working with composite functions?

A: You can practice working with composite functions by trying out different examples and exercises. You can also use online resources, such as calculators and software, to help you evaluate composite functions.

Q: What are some real-world applications of composite functions?

A: Some real-world applications of composite functions include:

Modeling the motion of objects in physics
Modeling the behavior of markets in economics
Evaluating the performance of complex systems in engineering

Q: Can I use composite functions to solve optimization problems?

A: Yes, composite functions can be used to solve optimization problems. For example, in economics, composite functions can be used to model the behavior of markets and optimize the allocation of resources.

Q: How do I know when to use a composite function to solve an optimization problem?

A: You should use a composite function to solve an optimization problem when you need to evaluate a function that takes another function as its input. This can be the case in a wide range of situations, including physics, engineering, and economics.

Q: What are some common pitfalls to avoid when using composite functions to solve optimization problems?

A: Some common pitfalls to avoid when using composite functions to solve optimization problems include:

Failing to evaluate the inner function before evaluating the composite function
Not simplifying the expression correctly
Not following the order of operations

Conclusion

In conclusion, composite functions are a powerful tool that can be used to solve a wide range of problems. By understanding how to evaluate composite functions and avoiding common mistakes, you can use composite functions to solve optimization problems and model complex systems.