Drag The Tiles To The Boxes To Form Correct Pairs.Consider Functions $f$ And $g$.$\[ \begin{align*} f(x) &= 1 - X^2 \\ g(x) &= \sqrt{11 - 4x} \end{align*} \\]Evaluate Each Combined Function, And Match It To The Corresponding

Introduction

In mathematics, combining functions is a fundamental concept that allows us to create new functions by combining existing ones. In this article, we will explore the concept of combining functions and evaluate two specific combined functions, $f \circ g$ and $g \circ f$, where $f(x) = 1 - x^2$ and $g(x) = \sqrt{11 - 4x}$. We will then match each combined function to its corresponding discussion category.

Understanding Function Composition

Before we dive into evaluating the combined functions, let's briefly discuss function composition. Function composition is the process of combining two or more functions to create a new function. The resulting function takes the output of one function as the input of another function. In mathematical notation, we denote function composition using the symbol $\circ$. For example, if we have two functions $f$ and $g$, the composition of $f$ and $g$ is denoted as $f \circ g$.

Evaluating $f \circ g$

To evaluate the combined function $f \circ g$, we need to substitute the expression for $g(x)$ into the expression for $f(x)$. This means we will replace $x$ in the expression for $f(x)$ with the expression for $g(x)$.

$$\begin{align*} (f \circ g)(x) &= f(g(x)) \\ &= 1 - (g(x))^2 \\ &= 1 - (\sqrt{11 - 4x})^2 \\ &= 1 - (11 - 4x) \\ &= -10 + 4x \end{align*}$$

Evaluating $g \circ f$

To evaluate the combined function $g \circ f$, we need to substitute the expression for $f(x)$ into the expression for $g(x)$. This means we will replace $x$ in the expression for $g(x)$ with the expression for $f(x)$.

$$\begin{align*} (g \circ f)(x) &= g(f(x)) \\ &= \sqrt{11 - 4f(x)} \\ &= \sqrt{11 - 4(1 - x^2)} \\ &= \sqrt{11 - 4 + 4x^2} \\ &= \sqrt{7 + 4x^2} \end{align*}$$

Matching Combined Functions to Discussion Categories

Now that we have evaluated the combined functions $f \circ g$ and $g \circ f$, we need to match each function to its corresponding discussion category. The discussion categories are:

• Algebra: This category includes functions that involve algebraic operations such as addition, subtraction, multiplication, and division.
• Trigonometry: This category includes functions that involve trigonometric operations such as sine, cosine, and tangent.
• Calculus: This category includes functions that involve calculus operations such as differentiation and integration.
• Statistics: This category includes functions that involve statistical operations such as mean, median, and standard deviation.

Based on the expressions for $f \circ g$ and $g \circ f$, we can match each function to its corresponding discussion category as follows:

• $f \circ g$: This function involves algebraic operations and can be matched to the Algebra discussion category.
• $g \circ f$: This function involves algebraic operations and can be matched to the Algebra discussion category.

Conclusion

In this article, we explored the concept of combining functions and evaluated two specific combined functions, $f \circ g$ and $g \circ f$. We then matched each combined function to its corresponding discussion category. The combined functions $f \circ g$ and $g \circ f$ can be matched to the Algebra discussion category. We hope this article has provided a clear understanding of function composition and its applications in mathematics.

Further Reading

For further reading on function composition and its applications in mathematics, we recommend the following resources:

• "Function Composition" by Khan Academy: This article provides a comprehensive introduction to function composition and its applications in mathematics.
• "Combining Functions" by Math Open Reference: This article provides a detailed explanation of function composition and its applications in mathematics.
• "Algebra" by Wolfram MathWorld: This article provides a comprehensive introduction to algebra and its applications in mathematics.

References

• "Calculus" by Michael Spivak: This book provides a comprehensive introduction to calculus and its applications in mathematics.
• "Algebra" by David S. Dummit and Richard M. Foote: This book provides a comprehensive introduction to algebra and its applications in mathematics.
• "Trigonometry" by Charles P. McKeague and Mark D. Turner: This book provides a comprehensive introduction to trigonometry and its applications in mathematics.
  

Introduction

In our previous article, we explored the concept of combining functions and evaluated two specific combined functions, $f \circ g$ and $g \circ f$, where $f(x) = 1 - x^2$ and $g(x) = \sqrt{11 - 4x}$. We then matched each combined function to its corresponding discussion category. In this article, we will answer some frequently asked questions related to function composition and its applications in mathematics.

Q&A

Q: What is function composition?

A: Function composition is the process of combining two or more functions to create a new function. The resulting function takes the output of one function as the input of another function.

Q: How do I evaluate a combined function?

A: To evaluate a combined function, you need to substitute the expression for one function into the expression for the other function. For example, to evaluate $f \circ g$, you need to substitute the expression for $g(x)$ into the expression for $f(x)$.

Q: What are some common applications of function composition?

A: Function composition has many applications in mathematics, including:

• Algebra: Function composition is used to solve equations and inequalities.
• Calculus: Function composition is used to find derivatives and integrals.
• Statistics: Function composition is used to analyze data and make predictions.

Q: Can I use function composition with any type of function?

A: No, function composition can only be used with functions that have the same domain and range. For example, you cannot compose a function with a function that has a different domain or range.

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

A: When composing two functions, you need to determine which function to compose first. This is usually determined by the order in which the functions are defined. For example, if you have two functions $f$ and $g$, you would typically compose $f$ with $g$ as $f \circ g$.

Q: Can I use function composition with trigonometric functions?

A: Yes, function composition can be used with trigonometric functions. For example, you can compose the sine function with the cosine function to create a new function.

Q: How do I use function composition to solve equations?

A: To use function composition to solve equations, you need to substitute the expression for one function into the expression for the other function. For example, to solve the equation $f(x) = g(x)$, you can substitute the expression for $g(x)$ into the expression for $f(x)$.

Conclusion

In this article, we answered some frequently asked questions related to function composition and its applications in mathematics. We hope this article has provided a clear understanding of function composition and its applications in mathematics.

Further Reading

For further reading on function composition and its applications in mathematics, we recommend the following resources:

• "Function Composition" by Khan Academy: This article provides a comprehensive introduction to function composition and its applications in mathematics.
• "Combining Functions" by Math Open Reference: This article provides a detailed explanation of function composition and its applications in mathematics.
• "Algebra" by Wolfram MathWorld: This article provides a comprehensive introduction to algebra and its applications in mathematics.

References

• "Calculus" by Michael Spivak: This book provides a comprehensive introduction to calculus and its applications in mathematics.
• "Algebra" by David S. Dummit and Richard M. Foote: This book provides a comprehensive introduction to algebra and its applications in mathematics.
• "Trigonometry" by Charles P. McKeague and Mark D. Turner: This book provides a comprehensive introduction to trigonometry and its applications in mathematics.