Given That $f(x)=3x-5$ And $g(x)=x^3-x$, What Is The Value Of \$f(g(2))$[/tex\]?A. 0 B. 1 C. 7 D. 13 E. 19

Introduction

In mathematics, composite functions are a fundamental concept that allows us to combine two or more functions to create a new function. In this article, we will explore how to evaluate composite functions, using the given functions $f(x)=3x-5$ and $g(x)=x^3-x$ as examples. We will specifically focus on finding the value of $f(g(2))$.

Understanding Composite Functions

A composite function is a function that is derived from two or more functions. It is denoted by $f(g(x))$, where $f(x)$ is the outer function and $g(x)$ is the inner function. To evaluate a composite function, we need to follow the order of operations, which is to evaluate the inner function first and then plug the result into the outer function.

Evaluating the Inner Function

In this case, we need to evaluate the inner function $g(x)=x^3-x$ at $x=2$. To do this, we simply substitute $x=2$ into the function and evaluate it.

$$g(2) = (2)^3 - (2)$$

$$g(2) = 8 - 2$$

$$g(2) = 6$$

Evaluating the Outer Function

Now that we have evaluated the inner function, we can plug the result into the outer function $f(x)=3x-5$. We will substitute $x=6$ into the function and evaluate it.

$$f(g(2)) = f(6)$$

$$f(g(2)) = 3(6) - 5$$

$$f(g(2)) = 18 - 5$$

$$f(g(2)) = 13$$

Conclusion

In this article, we have learned how to evaluate composite functions using the given functions $f(x)=3x-5$ and $g(x)=x^3-x$. We have specifically focused on finding the value of $f(g(2))$, which is equal to 13. By following the order of operations and evaluating the inner function first, we can easily evaluate composite functions and solve problems involving them.

Tips and Tricks

• When evaluating composite functions, always follow the order of operations and evaluate the inner function first.
• Make sure to substitute the correct value into the outer function.
• Use parentheses to group the terms and make the calculation easier.

Practice Problems

1. Evaluate the composite function $f(g(x))$, where $f(x)=2x+1$ and $g(x)=x^2-3$.
2. Find the value of $f(g(4))$, where $f(x)=x^2-2x+1$ and $g(x)=2x-1$.
3. Evaluate the composite function $f(g(x))$, where $f(x)=x^3-2x2+1$ and $g(x)=x+2$.

Solutions

1. $$f(g(x)) = f(x^2-3)$$

$$f(g(x)) = 2(x^2-3) + 1$$

$$f(g(x)) = 2x^2 - 6 + 1$$

$$f(g(x)) = 2x^2 - 5$$

2. $$f(g(4)) = f(2(4)-1)$$

$$f(g(4)) = f(8-1)$$

$$f(g(4)) = f(7)$$

$$f(g(4)) = 7^2 - 2(7) + 1$$

$$f(g(4)) = 49 - 14 + 1$$

$$f(g(4)) = 36$$

3. $$f(g(x)) = f(x+2)$$

$$f(g(x)) = (x+2)^3 - 2(x+2)^2 + 1$$

$$f(g(x)) = x^3 + 6x^2 + 12x + 8 - 2x^2 - 8x - 8 + 1$$

f(g(x)) = x^3 + 4x^2 + 4x + 1$<br/> **Evaluating Composite Functions: A Q&A Guide** ===================================================== **Introduction** --------------- In our previous article, we explored how to evaluate composite functions using the given functions $f(x)=3x-5$ and $g(x)=x^3-x$. We learned how to follow the order of operations and evaluate the inner function first. In this article, we will provide a Q&A guide to help you better understand composite functions and how to evaluate them. **Q: What is a composite function?** ----------------------------------- A: A composite function is a function that is derived from two or more functions. It is denoted by $f(g(x))$, where $f(x)$ is the outer function and $g(x)$ is the inner function. **Q: How do I evaluate a composite function?** --------------------------------------------- A: To evaluate a composite function, you need to follow the order of operations. First, evaluate the inner function $g(x)$ at the given value of $x$. Then, plug the result into the outer function $f(x)$ and evaluate it. **Q: What is the order of operations for evaluating composite functions?** ------------------------------------------------------------------- A: The order of operations for evaluating composite functions is as follows: 1. Evaluate the inner function $g(x)$ at the given value of $x$. 2. Plug the result into the outer function $f(x)$. 3. Evaluate the outer function $f(x)$. **Q: How do I know which function is the inner function and which is the outer function?** ----------------------------------------------------------------------------------- A: The inner function is the function that is evaluated first, and the outer function is the function that is evaluated second. In the notation $f(g(x))$, $g(x)$ is the inner function and $f(x)$ is the outer function. **Q: Can I evaluate composite functions with more than two functions?** ------------------------------------------------------------------- A: Yes, you can evaluate composite functions with more than two functions. For example, if you have three functions $f(x)$, $g(x)$, and $h(x)$, you can evaluate the composite function $f(g(h(x)))$. **Q: How do I handle parentheses when evaluating composite functions?** ------------------------------------------------------------------- A: When evaluating composite functions, you need to follow the order of operations and use parentheses to group the terms. For example, if you have the composite function $f(g(x)) = f(x^2-3)$, you need to evaluate the inner function $g(x)$ first and then plug the result into the outer function $f(x)$. **Q: Can I use composite functions to solve real-world problems?** ---------------------------------------------------------------- A: Yes, composite functions can be used to solve real-world problems. For example, if you are modeling the growth of a population, you may need to use composite functions to evaluate the population at different times. **Q: How do I practice evaluating composite functions?** --------------------------------------------------- A: You can practice evaluating composite functions by working through examples and exercises. You can also use online resources and practice problems to help you improve your skills. **Practice Problems** ------------------- 1. Evaluate the composite function $f(g(x))$, where $f(x)=2x+1$ and $g(x)=x^2-3$. 2. Find the value of $f(g(4))$, where $f(x)=x^2-2x+1$ and $g(x)=2x-1$. 3. Evaluate the composite function $f(g(x))$, where $f(x)=x^3-2x^2+1$ and $g(x)=x+2$. **Solutions** ------------ 1. $f(g(x)) = f(x^2-3)

$$f(g(x)) = 2(x^2-3) + 1$$

$$f(g(x)) = 2x^2 - 6 + 1$$

$$f(g(x)) = 2x^2 - 5$$

2. $$f(g(4)) = f(2(4)-1)$$

$$f(g(4)) = f(8-1)$$

$$f(g(4)) = f(7)$$

$$f(g(4)) = 7^2 - 2(7) + 1$$

$$f(g(4)) = 49 - 14 + 1$$

$$f(g(4)) = 36$$

3. $$f(g(x)) = f(x+2)$$

$$f(g(x)) = (x+2)^3 - 2(x+2)^2 + 1$$

$$f(g(x)) = x^3 + 6x^2 + 12x + 8 - 2x^2 - 8x - 8 + 1$$

$$f(g(x)) = x^3 + 4x^2 + 4x + 1$$

Conclusion

In this article, we have provided a Q&A guide to help you better understand composite functions and how to evaluate them. We have covered topics such as the order of operations, how to handle parentheses, and how to practice evaluating composite functions. By following the steps outlined in this article, you can become proficient in evaluating composite functions and solving real-world problems.