Given The Following Table With Selected Values Of The Functions $f(x$\] And $g(x$\], Determine $f(g(2)) - G(f(-1)$\].$\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -5 & -4 & -1 & 2 & 4 & 7 \\ \hline f(x) & 21 & 17 & -1 & -7 &

Introduction

In mathematics, composite functions are a fundamental concept that allows us to combine two or more functions to create a new function. Given a table with selected values of the functions $f(x)$ and $g(x)$, we are tasked with determining the value of $f(g(2)) - g(f(-1))$. In this article, we will break down the problem step by step and provide a clear explanation of the process.

Understanding Composite Functions

Before we dive into the problem, let's take a moment to understand what composite functions are. A composite function is a function that is defined in terms of another function. In other words, we take the output of one function and use it as the input for another function. The notation $f(g(x))$ represents a composite function where the output of $g(x)$ is used as the input for $f(x)$.

Given Table

We are given a table with selected values of the functions $f(x)$ and $g(x)$.

Step 1: Evaluate $g(2)$

To evaluate $g(2)$, we need to find the value of $g(x)$ when $x = 2$. From the table, we can see that $g(2) = 2$.

Step 2: Evaluate $f(g(2))$

Now that we have the value of $g(2)$, we can evaluate $f(g(2))$. Since $g(2) = 2$, we need to find the value of $f(x)$ when $x = 2$. From the table, we can see that $f(2) = -7$. Therefore, $f(g(2)) = f(2) = -7$.

Step 3: Evaluate $f(-1)$

To evaluate $f(-1)$, we need to find the value of $f(x)$ when $x = -1$. From the table, we can see that $f(-1) = -1$.

Step 4: Evaluate $g(f(-1))$

Now that we have the value of $f(-1)$, we can evaluate $g(f(-1))$. Since $f(-1) = -1$, we need to find the value of $g(x)$ when $x = -1$. From the table, we can see that $g(-1) = -1$. Therefore, $g(f(-1)) = g(-1) = -1$.

Step 5: Evaluate $f(g(2)) - g(f(-1))$

Finally, we can evaluate $f(g(2)) - g(f(-1))$. We have already found the values of $f(g(2))$ and $g(f(-1))$ in the previous steps. Therefore, $f(g(2)) - g(f(-1)) = -7 - (-1) = -7 + 1 = -6$.

Conclusion

Q&A: Evaluating Composite Functions

Q: What is a composite function?

A: A composite function is a function that is defined in terms of another function. In other words, we take the output of one function and use it as the input for another function. The notation $f(g(x))$ represents a composite function where the output of $g(x)$ is used as the input for $f(x)$.

Q: How do I evaluate a composite function?

A: To evaluate a composite function, you need to follow these steps:

1. Evaluate the inner function (in this case, $g(x)$).
2. Use the output of the inner function as the input for the outer function (in this case, $f(x)$).
3. Evaluate the outer function using the input from step 2.

Q: What if the table does not have the value I need?

A: If the table does not have the value you need, you cannot evaluate the composite function using the table. However, you can try to find the value using other methods, such as graphing or using a calculator.

Q: Can I use a composite function with more than two functions?

A: Yes, you can use a composite function with more than two functions. For example, you can use the notation $f(g(h(x)))$ to represent a composite function where the output of $h(x)$ is used as the input for $g(x)$, and the output of $g(x)$ is used as the input for $f(x)$.

Q: How do I know which function to evaluate first?

A: To determine which function to evaluate first, you need to look at the notation of the composite function. The function that is inside the parentheses is evaluated first. For example, in the notation $f(g(x))$, the function $g(x)$ is evaluated first.

Q: Can I use a composite function with different types of functions?

A: Yes, you can use a composite function with different types of functions. For example, you can use a composite function with a linear function and a quadratic function.

Q: How do I evaluate a composite function with a negative input?

A: To evaluate a composite function with a negative input, you need to follow the same steps as before. However, you need to be careful when evaluating the inner function, as the output may be negative.

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

A: Yes, you can use a composite function to solve real-world problems. For example, you can use a composite function to model the growth of a population, where the output of one function is used as the input for another function.

Conclusion

In this article, we have answered some common questions about evaluating composite functions. We have provided a step-by-step guide on how to evaluate a composite function, and we have discussed some common pitfalls to avoid. We hope that this article has been helpful in understanding composite functions and how to evaluate them.

Additional Resources

• Mathway: A online math problem solver that can help you evaluate composite functions.
• Khan Academy: A online learning platform that provides video lessons and practice exercises on composite functions.
• Wolfram Alpha: A online calculator that can help you evaluate composite functions and other mathematical expressions.