Given The Functions $f(x)=x+\frac{1}{x}$ And $g(x)=\frac{x+1}{x+2}$, Find The Function $g \circ F$ And The Domain Of $g \circ F$.

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Introduction

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$, the composition of $g$ and $f$ is denoted by $g \circ f$ and is defined as $(g \circ f)(x) = g(f(x))$. In this article, we will explore the composition of the functions $f(x)=x+\frac{1}{x}$ and $g(x)=\frac{x+1}{x+2}$, and find the function $g \circ f$ and its domain.

The Function $f(x)$

The function $f(x)$ is defined as $f(x)=x+\frac{1}{x}$. This function is a rational function, which means it is the ratio of two polynomials. The domain of $f(x)$ is all real numbers except for $x=0$, since division by zero is undefined.

The Function $g(x)$

The function $g(x)$ is defined as $g(x)=\frac{x+1}{x+2}$. This function is also a rational function, and its domain is all real numbers except for $x=-2$, since division by zero is undefined.

Finding the Composition $g \circ f$

To find the composition $g \circ f$, we need to substitute $f(x)$ into $g(x)$ in place of $x$. This means we will replace every instance of $x$ in the expression for $g(x)$ with the expression for $f(x)$.

$(g \circ f)(x) = g(f(x)) = \frac{f(x)+1}{f(x)+2} = \frac{\left(x+\frac{1}{x}\right)+1}{\left(x+\frac{1}{x}\right)+2}$

To simplify this expression, we can start by combining the terms in the numerator and denominator.

$(g \circ f)(x) = \frac{x+\frac{1}{x}+1}{x+\frac{1}{x}+2}$

We can rewrite the numerator and denominator as a single fraction by finding a common denominator.

$(g \circ f)(x) = \frac{x+\frac{1}{x}+\frac{2}{2}}{x+\frac{1}{x}+\frac{4}{2}} = \frac{\frac{2x^2+2+x}{2}}{\frac{2x^2+4+x}{2}}$

Now we can simplify the expression by canceling out the common factors in the numerator and denominator.

$(g \circ f)(x) = \frac{2x^2+2+x}{2x^2+4+x}$

Simplifying the Composition

We can simplify the composition $g \circ f$ further by factoring the numerator and denominator.

$(g \circ f)(x) = \frac{(2x+1)(x+1)}{(2x+2)(x+2)}$

Now we can cancel out the common factors in the numerator and denominator.

$(g \circ f)(x) = \frac{x+1}{x+2}$

Finding the Domain of $g \circ f$

The domain of $g \circ f$ is the set of all real numbers for which the composition is defined. Since the composition is a rational function, it is defined for all real numbers except for those that make the denominator zero.

To find the values of $x$ that make the denominator zero, we can set the denominator equal to zero and solve for $x$.

$x+2=0 \Rightarrow x=-2$

Therefore, the domain of $g \circ f$ is all real numbers except for $x=-2$.

Conclusion

In this article, we found the composition $g \circ f$ of the functions $f(x)=x+\frac{1}{x}$ and $g(x)=\frac{x+1}{x+2}$, and determined its domain. We showed that the composition is a rational function, and that its domain is all real numbers except for $x=-2$. This demonstrates the importance of understanding the composition of functions and their domains in mathematics.

References

• [1] "Composition of Functions" by Math Open Reference
• [2] "Rational Functions" by Math Is Fun

Further Reading

• [1] "Functions" by Khan Academy
• [2] "Composition of Functions" by Wolfram MathWorld
  Q&A: Composition of Functions and its Domain =====================================================

Introduction

In our previous article, we explored the composition of the functions $f(x)=x+\frac{1}{x}$ and $g(x)=\frac{x+1}{x+2}$, and found the function $g \circ f$ and its domain. In this article, we will answer some frequently asked questions about composition of functions and its domain.

Q: What is the composition of functions?

A: The composition of functions is a way of combining two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$, the composition of $g$ and $f$ is denoted by $g \circ f$ and is defined as $(g \circ f)(x) = g(f(x))$.

Q: How do I find the composition of two functions?

A: To find the composition of two functions, you need to substitute one function into the other in place of the variable. For example, if we want to find the composition of $f(x)=x+\frac{1}{x}$ and $g(x)=\frac{x+1}{x+2}$, we would substitute $f(x)$ into $g(x)$ in place of $x$.

Q: What is the domain of a composition of functions?

A: The domain of a composition of functions is the set of all real numbers for which the composition is defined. Since the composition is a rational function, it is defined for all real numbers except for those that make the denominator zero.

Q: How do I find the domain of a composition of functions?

A: To find the domain of a composition of functions, you need to find the values of $x$ that make the denominator zero, and exclude those values from the domain.

Q: What is the difference between the composition of functions and the product of functions?

A: The composition of functions is a way of combining two or more functions to create a new function, whereas the product of functions is a way of multiplying two or more functions together. For example, if we have two functions $f(x)$ and $g(x)$, the composition of $g$ and $f$ is denoted by $g \circ f$, whereas the product of $f$ and $g$ is denoted by $f \cdot g$.

Q: Can I use the composition of functions to solve equations?

A: Yes, you can use the composition of functions to solve equations. For example, if we have an equation of the form $g(f(x)) = k$, where $k$ is a constant, we can use the composition of functions to solve for $x$.

Q: What are some common applications of the composition of functions?

A: The composition of functions has many common applications in mathematics and science, including:

• Modeling real-world phenomena, such as population growth and chemical reactions
• Solving equations and inequalities
• Finding the inverse of a function
• Graphing functions

Conclusion

In this article, we answered some frequently asked questions about composition of functions and its domain. We hope that this article has been helpful in clarifying some of the concepts related to composition of functions and its domain.

References

• [1] "Composition of Functions" by Math Open Reference
• [2] "Rational Functions" by Math Is Fun

Further Reading

• [1] "Functions" by Khan Academy
• [2] "Composition of Functions" by Wolfram MathWorld