For The Real-valued Functions $f(x)=\sqrt{2x+18}$ And $g(x)=x-5$, Find The Composition $f \circ G$ And Specify Its Domain Using Interval Notation.$(f \circ G)(x) = $Domain Of $f \circ G$: $\square$

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 real-valued functions $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $f \circ g$ and is defined as $(f \circ g)(x) = f(g(x))$. In this article, we will explore the composition of the functions $f(x)=\sqrt{2x+18}$ and $g(x)=x-5$, and determine its domain using interval notation.

Composition of Functions

To find the composition $f \circ g$, we need to substitute $g(x)$ into $f(x)$ in place of $x$. This means that we will replace every instance of $x$ in the function $f(x)$ with the expression $g(x) = x - 5$. The resulting function is:

$$(f \circ g)(x) = f(g(x)) = \sqrt{2(x-5)+18}$$

Simplifying the expression inside the square root, we get:

$$(f \circ g)(x) = \sqrt{2x-10+18} = \sqrt{2x+8}$$

Domain Analysis

To determine the domain of the composition $f \circ g$, we need to consider the restrictions imposed by both functions $f(x)$ and $g(x)$. The function $f(x) = \sqrt{2x+18}$ is defined only when the expression inside the square root is non-negative, i.e., $2x+18 \geq 0$. Solving this inequality, we get:

$$2x+18 \geq 0 \Rightarrow x \geq -9$$

The function $g(x) = x-5$ is defined for all real numbers $x$. However, since the composition $f \circ g$ is defined only when the expression inside the square root is non-negative, we need to ensure that $2x+8 \geq 0$. Solving this inequality, we get:

$$2x+8 \geq 0 \Rightarrow x \geq -4$$

Since the composition $f \circ g$ is defined only when both $x \geq -9$ and $x \geq -4$, we can conclude that the domain of $f \circ g$ is the intersection of these two intervals:

$$\text{Domain of } f \circ g = [-4, \infty)$$

Conclusion

In this article, we have explored the composition of the functions $f(x)=\sqrt{2x+18}$ and $g(x)=x-5$, and determined its domain using interval notation. We have shown that the composition $f \circ g$ is defined as $(f \circ g)(x) = \sqrt{2x+8}$, and that its domain is the interval $[-4, \infty)$. This analysis demonstrates the importance of considering the restrictions imposed by both functions in the composition, and highlights the need for careful domain analysis when working with composite functions.

Key Takeaways

• The composition of functions is a fundamental concept in mathematics that allows us to combine two or more functions to create a new function.
• To find the composition of two functions, we need to substitute one function into the other in place of the variable.
• The domain of a composite function is determined by the restrictions imposed by both functions in the composition.
• Careful domain analysis is essential when working with composite functions to ensure that the resulting function is well-defined and meaningful.

Further Reading

For those interested in exploring further, we recommend the following resources:

• Wikipedia: Composition of functions
• Khan Academy: Composition of functions
• Mathway: Composition of functions
  Composition of Functions: Q&A =============================

Introduction

In our previous article, we explored the composition of the functions $f(x)=\sqrt{2x+18}$ and $g(x)=x-5$, and determined its domain using interval notation. In this article, we will address some common questions and concerns related to the composition of functions.

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 $f$ and $g$ is denoted by $f \circ g$ and is defined as $(f \circ g)(x) = f(g(x))$.

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 $f(x)=\sqrt{2x+18}$ and $g(x)=x-5$, we would substitute $g(x)$ into $f(x)$ in place of $x$.

Q: What is the domain of a composite function?

A: The domain of a composite function is determined by the restrictions imposed by both functions in the composition. In other words, the domain of the composite function is the intersection of the domains of the two individual functions.

Q: How do I determine the domain of a composite function?

A: To determine the domain of a composite function, you need to consider the restrictions imposed by both functions in the composition. For example, if we have a composite function $f \circ g$ where $f(x)$ is defined only when $x \geq -9$ and $g(x)$ is defined for all real numbers $x$, then the domain of $f \circ g$ would be the intersection of these two intervals.

Q: What are some common mistakes to avoid when working with composite functions?

A: Some common mistakes to avoid when working with composite functions include:

• Failing to consider the restrictions imposed by both functions in the composition.
• Not properly substituting one function into the other in place of the variable.
• Not checking the domain of the composite function to ensure that it is well-defined and meaningful.

Q: How do I graph a composite function?

A: Graphing a composite function can be a bit more challenging than graphing a single function. However, there are a few tips that can help:

• Start by graphing the inner function $g(x)$.
• Then, substitute the expression for $g(x)$ into the outer function $f(x)$.
• Graph the resulting function, making sure to consider the restrictions imposed by both functions in the composition.

Q: What are some real-world applications of composite functions?

A: Composite functions have many real-world applications, including:

• Modeling population growth and decline.
• Analyzing financial data and making predictions about future trends.
• Understanding complex systems and making predictions about their behavior.

Conclusion

In this article, we have addressed some common questions and concerns related to the composition of functions. We hope that this Q&A has been helpful in clarifying some of the concepts and providing additional insights into the world of composite functions.

Key Takeaways

• The composition of functions is a way of combining two or more functions to create a new function.
• The domain of a composite function is determined by the restrictions imposed by both functions in the composition.
• Graphing a composite function can be a bit more challenging than graphing a single function.
• Composite functions have many real-world applications, including modeling population growth and decline, analyzing financial data, and understanding complex systems.

Further Reading

For those interested in exploring further, we recommend the following resources:

• Wikipedia: Composition of functions
• Khan Academy: Composition of functions
• Mathway: Composition of functions