Express Your Answer As A Polynomial In Standard Form.Given: $\[ \begin{array}{l} f(x) = 2x^2 - 6x + 15 \\ g(x) = -2x - 4 \end{array} \\]Find: \[$(f \circ G)(x)\$\]

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Introduction

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In mathematics, composite functions are a fundamental concept that allows us to combine multiple functions to create new functions. When working with composite functions, it's often necessary to express them in standard form, which is a polynomial expression. In this article, we will explore how to express composite functions as polynomials in standard form using the given functions $f(x) = 2x^2 - 6x + 15$ and $g(x) = -2x - 4$.

Understanding Composite Functions

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A composite function is a function that is derived from the composition of two or more functions. In this case, we are given two functions, $f(x)$ and $g(x)$, and we need to find the composite function $(f \circ g)(x)$. The composite function $(f \circ g)(x)$ is defined as $f(g(x))$, which means that we need to plug in the expression for $g(x)$ into the function $f(x)$.

Finding the Composite Function

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To find the composite function $(f \circ g)(x)$, we need to substitute the expression for $g(x)$ into the function $f(x)$. This means that we will replace $x$ in the function $f(x)$ with the expression $g(x) = -2x - 4$.

Step 1: Substitute the expression for g(x) into f(x)

We will start by substituting the expression for $g(x)$ into the function $f(x)$. This gives us:

$f(g(x)) = 2(-2x - 4)^2 - 6(-2x - 4) + 15$

Step 2: Expand the expression

Next, we need to expand the expression for $f(g(x))$. This involves multiplying out the squared term and simplifying the expression.

$f(g(x)) = 2(-2x - 4)^2 - 6(-2x - 4) + 15$

$f(g(x)) = 2(4x^2 + 16x + 16) + 12x + 24 + 15$

$f(g(x)) = 8x^2 + 32x + 32 + 12x + 24 + 15$

$f(g(x)) = 8x^2 + 44x + 71$

Step 3: Simplify the expression

Finally, we need to simplify the expression for $f(g(x))$. This involves combining like terms and rearranging the expression in standard form.

$f(g(x)) = 8x^2 + 44x + 71$

Conclusion

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In this article, we have shown how to express composite functions as polynomials in standard form using the given functions $f(x) = 2x^2 - 6x + 15$ and $g(x) = -2x - 4$. We have demonstrated the step-by-step process of finding the composite function $(f \circ g)(x)$ and have simplified the expression to obtain the final answer in standard form.

Final Answer

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The final answer to the problem is:

$(f \circ g)(x) = 8x^2 + 44x + 71$

This is the polynomial expression for the composite function $(f \circ g)(x)$ in standard form.

Discussion

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The concept of composite functions is a fundamental idea in mathematics that allows us to combine multiple functions to create new functions. When working with composite functions, it's often necessary to express them in standard form, which is a polynomial expression. In this article, we have demonstrated the step-by-step process of finding the composite function $(f \circ g)(x)$ and have simplified the expression to obtain the final answer in standard form.

Key Takeaways

• Composite functions are a fundamental concept in mathematics that allows us to combine multiple functions to create new functions.
• When working with composite functions, it's often necessary to express them in standard form, which is a polynomial expression.
• To find the composite function $(f \circ g)(x)$, we need to substitute the expression for $g(x)$ into the function $f(x)$.
• We can simplify the expression for $f(g(x))$ by combining like terms and rearranging the expression in standard form.

Future Directions

• In future articles, we can explore more advanced topics in mathematics, such as calculus and differential equations.
• We can also discuss the applications of composite functions in real-world scenarios, such as physics and engineering.
• Additionally, we can explore the concept of inverse functions and how they relate to composite functions.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Differential Equations" by Lawrence Perko

Glossary

• Composite function: A function that is derived from the composition of two or more functions.
• Standard form: A polynomial expression that is written in the form $ax^2 + bx + c$.
• Polynomial expression: An expression that is written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.
  

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Introduction

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In our previous article, we explored how to express composite functions as polynomials in standard form using the given functions $f(x) = 2x^2 - 6x + 15$ and $g(x) = -2x - 4$. In this article, we will answer some frequently asked questions (FAQs) related to expressing composite functions as polynomials in standard form.

Q&A

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Q: What is a composite function?

A: A composite function is a function that is derived from the composition of two or more functions. In this case, we are given two functions, $f(x)$ and $g(x)$, and we need to find the composite function $(f \circ g)(x)$.

Q: How do I find the composite function $(f \circ g)(x)$?

A: To find the composite function $(f \circ g)(x)$, we need to substitute the expression for $g(x)$ into the function $f(x)$. This means that we will replace $x$ in the function $f(x)$ with the expression $g(x) = -2x - 4$.

Q: What is the standard form of a polynomial expression?

A: The standard form of a polynomial expression is written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Q: How do I simplify the expression for $f(g(x))$?

A: To simplify the expression for $f(g(x))$, we need to combine like terms and rearrange the expression in standard form. This involves multiplying out the squared term and simplifying the expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

$(f \circ g)(x) = 8x^2 + 44x + 71$

This is the polynomial expression for the composite function $(f \circ g)(x)$ in standard form.

Q: Can I use this method to find the composite function $(f \circ g)(x)$ for any two functions $f(x)$ and $g(x)$?

A: Yes, you can use this method to find the composite function $(f \circ g)(x)$ for any two functions $f(x)$ and $g(x)$. However, you need to make sure that the functions are defined and that the composite function is well-defined.

Q: What are some common mistakes to avoid when finding the composite function $(f \circ g)(x)$?

A: Some common mistakes to avoid when finding the composite function $(f \circ g)(x)$ include:

• Not substituting the expression for $g(x)$ into the function $f(x)$ correctly.
• Not simplifying the expression for $f(g(x))$ correctly.
• Not checking if the composite function is well-defined.

Conclusion

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In this article, we have answered some frequently asked questions (FAQs) related to expressing composite functions as polynomials in standard form. We have also provided some common mistakes to avoid when finding the composite function $(f \circ g)(x)$. By following the steps outlined in this article, you should be able to express composite functions as polynomials in standard form with ease.

Final Answer

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The final answer to the problem is:

$(f \circ g)(x) = 8x^2 + 44x + 71$

This is the polynomial expression for the composite function $(f \circ g)(x)$ in standard form.

Discussion

---

The concept of composite functions is a fundamental idea in mathematics that allows us to combine multiple functions to create new functions. When working with composite functions, it's often necessary to express them in standard form, which is a polynomial expression. In this article, we have demonstrated the step-by-step process of finding the composite function $(f \circ g)(x)$ and have simplified the expression to obtain the final answer in standard form.

Key Takeaways

• Composite functions are a fundamental concept in mathematics that allows us to combine multiple functions to create new functions.
• When working with composite functions, it's often necessary to express them in standard form, which is a polynomial expression.
• To find the composite function $(f \circ g)(x)$, we need to substitute the expression for $g(x)$ into the function $f(x)$.
• We can simplify the expression for $f(g(x))$ by combining like terms and rearranging the expression in standard form.

Future Directions

• In future articles, we can explore more advanced topics in mathematics, such as calculus and differential equations.
• We can also discuss the applications of composite functions in real-world scenarios, such as physics and engineering.
• Additionally, we can explore the concept of inverse functions and how they relate to composite functions.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Differential Equations" by Lawrence Perko

Glossary

• Composite function: A function that is derived from the composition of two or more functions.
• Standard form: A polynomial expression that is written in the form $ax^2 + bx + c$.
• Polynomial expression: An expression that is written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.