Question 3. [6, 6, 6, 6 Marks]Consider The Following Functions: $\[ F(x) = X + \frac{1}{x} \\] $\[ G(x) = \frac{x+1}{x+2} \\](a) Find The Function \[$ F \circ G \$\] And The Domain \[$ D_{f \circ G} \$\].(b) Find The
Introduction
In this problem, we are given two functions, and , and we are asked to find the composition of these functions, denoted as , and determine its domain, . The composition of functions is a fundamental concept in mathematics, and it plays a crucial role in various fields, including calculus, algebra, and analysis.
Function Definitions
The two functions given are:
Composition of Functions
To find the composition of functions and , we need to substitute the expression for into the function . This means that we will replace in the function with the expression for .
Now, we substitute the expression for into the above equation:
To simplify the above expression, we can multiply both the numerator and denominator of the second term by :
Now, we can combine the two fractions by finding a common denominator, which is :
Expanding the numerator, we get:
Simplifying the numerator, we get:
Domain of the Composition
To determine the domain of the composition , we need to find the values of for which the function is defined. In other words, we need to find the values of for which the denominator of the function is not equal to zero.
Looking at the denominator of the function , we can see that it is a quadratic expression:
To find the values of for which this expression is not equal to zero, we can factor the quadratic expression:
Now, we can see that the expression is not equal to zero when is not equal to or . Therefore, the domain of the composition is:
Conclusion
In this problem, we found the composition of the functions and , denoted as , and determined its domain, . The composition of functions is a fundamental concept in mathematics, and it plays a crucial role in various fields, including calculus, algebra, and analysis.
Final Answer
The final answer is:
D_{f \circ g} = (-\infty, -2) \cup (-2, -1) \cup (-1, \infty)$<br/> **Q&A: Composition of Functions and Domain Analysis** ===================================================== **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. This is done by substituting the expression for one function into the other function. **Q: How do you find the composition of two functions?** ------------------------------------------------ A: To find the composition of two functions, you need to substitute the expression for one function into the other function. This means that you will replace the variable in the second function with the expression for the first function. **Q: What is the domain of a composition of functions?** --------------------------------------------------- A: The domain of a composition of functions is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is not equal to zero. **Q: How do you determine the domain of a composition of functions?** ---------------------------------------------------------------- A: To determine the domain of a composition of functions, you need to find the values of x for which the denominator of the function is not equal to zero. This means that you need to find the values of x for which the expression in the denominator is not equal to zero. **Q: What is the difference between the domain and the range of a function?** ------------------------------------------------------------------- A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values for which the function is defined. **Q: How do you find the range of a composition of functions?** --------------------------------------------------------- A: To find the range of a composition of functions, you need to find the set of all possible output values for which the function is defined. This means that you need to find the set of all possible values of the function for which the input values are in the domain of the function. **Q: What is the significance of the composition of functions in real-world applications?** -------------------------------------------------------------------------------- A: The composition of functions has many real-world applications, including: * Modeling population growth and decline * Analyzing the behavior of complex systems * Solving optimization problems * Creating mathematical models for real-world phenomena **Q: How do you use the composition of functions to solve optimization problems?** ------------------------------------------------------------------------- A: To use the composition of functions to solve optimization problems, you need to find the maximum or minimum value of a function. This can be done by finding the critical points of the function and then using the second derivative test to determine whether the critical point is a maximum or minimum. **Q: What are some common mistakes to avoid when working with the composition of functions?** ----------------------------------------------------------------------------------- A: Some common mistakes to avoid when working with the composition of functions include: * Not checking the domain of the function * Not simplifying the expression for the composition of functions * Not using the correct notation for the composition of functions **Q: How do you check the domain of a composition of functions?** --------------------------------------------------------- A: To check the domain of a composition of functions, you need to find the values of x for which the denominator of the function is not equal to zero. This means that you need to find the values of x for which the expression in the denominator is not equal to zero. **Q: What is the importance of the composition of functions in calculus?** ------------------------------------------------------------------- A: The composition of functions is a fundamental concept in calculus, and it plays a crucial role in many areas of calculus, including: * Differentiation * Integration * Optimization * Mathematical modeling **Q: How do you use the composition of functions to solve differentiation problems?** ------------------------------------------------------------------------- A: To use the composition of functions to solve differentiation problems, you need to find the derivative of a function. This can be done by using the chain rule, which states that the derivative of a composite function is the product of the derivatives of the individual functions. **Q: What are some common applications of the composition of functions in real-world scenarios?** ----------------------------------------------------------------------------------- A: Some common applications of the composition of functions in real-world scenarios include: * Modeling population growth and decline * Analyzing the behavior of complex systems * Solving optimization problems * Creating mathematical models for real-world phenomena **Q: How do you use the composition of functions to solve integration problems?** ------------------------------------------------------------------------- A: To use the composition of functions to solve integration problems, you need to find the integral of a function. This can be done by using the fundamental theorem of calculus, which states that the integral of a function is equal to the antiderivative of the function.