For \[$ F(x)=\frac{7}{x-9} \$\] And \[$ G(x)=\frac{3}{x} \$\], Find The Following Composite Functions And State The Domain Of Each.(a) \[$ F \circ G \$\](b) \[$ G \circ F \$\]$\[ (g \circ F)(x)=\frac{3x-27}{7}

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Introduction

In mathematics, composite functions are 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 composite function (f ∘ g)(x) is defined as f(g(x)). In this article, we will explore the composite functions of f(x) = 7/(x-9) and g(x) = 3/x, and analyze their domains.

Function Definitions

f(x) = 7/(x-9)

The function f(x) is defined as f(x) = 7/(x-9). This function has a vertical asymptote at x = 9, which means that the function is undefined at x = 9.

g(x) = 3/x

The function g(x) is defined as g(x) = 3/x. This function has a vertical asymptote at x = 0, which means that the function is undefined at x = 0.

Composite Functions

(a) f ∘ g

To find the composite function (f ∘ g)(x), we need to substitute g(x) into f(x).

(f ∘ g)(x) = f(g(x)) = f(3/x) = 7/(3/x - 9)

To simplify this expression, we can multiply the numerator and denominator by x.

(f ∘ g)(x) = 7x / (3 - 9x)

(b) g ∘ f

To find the composite function (g ∘ f)(x), we need to substitute f(x) into g(x).

(g ∘ f)(x) = g(f(x)) = g(7/(x-9)) = 3 / (7/(x-9))

To simplify this expression, we can multiply the numerator and denominator by (x-9).

(g ∘ f)(x) = 3(x-9) / 7

Simplifying the Composite Functions

We can simplify the composite functions further by combining like terms.

(f ∘ g)(x) = 7x / (3 - 9x)

(g ∘ f)(x) = (3x - 27) / 7

Domain Analysis

Domain of f(x)

The domain of f(x) is all real numbers except x = 9, since the function is undefined at x = 9.

Domain of g(x)

The domain of g(x) is all real numbers except x = 0, since the function is undefined at x = 0.

Domain of (f ∘ g)(x)

To find the domain of (f ∘ g)(x), we need to consider the values of x that make the denominator of the composite function equal to zero.

(f ∘ g)(x) = 7x / (3 - 9x)

The denominator is equal to zero when 3 - 9x = 0.

3 - 9x = 0

-9x = -3

x = 1/3

Therefore, the domain of (f ∘ g)(x) is all real numbers except x = 1/3.

Domain of (g ∘ f)(x)

To find the domain of (g ∘ f)(x), we need to consider the values of x that make the denominator of the composite function equal to zero.

(g ∘ f)(x) = (3x - 27) / 7

The denominator is equal to zero when 7 = 0, which is not possible.

Therefore, the domain of (g ∘ f)(x) is all real numbers.

Conclusion

Introduction

In our previous article, we explored the composite functions of f(x) = 7/(x-9) and g(x) = 3/x, and analyzed their domains. In this article, we will answer some frequently asked questions about composite functions and domain analysis.

Q&A

Q: What is a composite function?

A: A composite function is a function that is formed by combining two or more functions. Given two functions, f(x) and g(x), the composite function (f ∘ g)(x) is defined as f(g(x)).

Q: How do I find the composite function (f ∘ g)(x)?

A: To find the composite function (f ∘ g)(x), you need to substitute g(x) into f(x). For example, if f(x) = 7/(x-9) and g(x) = 3/x, then (f ∘ g)(x) = f(g(x)) = f(3/x) = 7/(3/x - 9).

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all possible input values for which the composite function is defined. In other words, it is the set of all values of x for which the composite function is not undefined.

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

A: To find the domain of a composite function, you need to consider the values of x that make the denominator of the composite function equal to zero. You also need to consider any values of x that make the composite function undefined.

Q: What is the difference between the domain of f(x) and the domain of (f ∘ g)(x)?

A: The domain of f(x) is the set of all possible input values for which f(x) is defined. The domain of (f ∘ g)(x) is the set of all possible input values for which (f ∘ g)(x) is defined. In general, the domain of (f ∘ g)(x) is a subset of the domain of f(x).

Q: Can I always find the composite function (f ∘ g)(x)?

A: No, you cannot always find the composite function (f ∘ g)(x). If the composite function is undefined for some value of x, then you cannot find the composite function.

Q: Can I always find the domain of a composite function?

A: No, you cannot always find the domain of a composite function. If the composite function is undefined for some value of x, then you cannot find the domain of the composite function.

Q: How do I simplify a composite function?

A: To simplify a composite function, you can use algebraic manipulations such as combining like terms, canceling out common factors, and rearranging the expression.

Q: What is the importance of composite functions in mathematics?

A: Composite functions are important in mathematics because they allow us to combine two or more functions to create a new function. This is useful in a variety of applications, including physics, engineering, and economics.

Conclusion

In this article, we have answered some frequently asked questions about composite functions and domain analysis. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in composite functions and domain analysis.

Common Mistakes to Avoid

  • Not considering the domain of the composite function
  • Not simplifying the composite function
  • Not using algebraic manipulations to simplify the composite function
  • Not checking for undefined values of x

Tips and Tricks

  • Always check the domain of the composite function before finding the composite function.
  • Use algebraic manipulations to simplify the composite function.
  • Check for undefined values of x before finding the composite function.
  • Use a calculator or computer software to check the domain of the composite function.

Practice Problems

  • Find the composite function (f ∘ g)(x) given f(x) = 7/(x-9) and g(x) = 3/x.
  • Find the domain of the composite function (f ∘ g)(x) given f(x) = 7/(x-9) and g(x) = 3/x.
  • Simplify the composite function (f ∘ g)(x) given f(x) = 7/(x-9) and g(x) = 3/x.

Conclusion

In this article, we have provided some practice problems to help you practice finding composite functions and their domains. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in composite functions and domain analysis.