Suppose That The Functions { G $}$ And { H $}$ Are Defined As Follows:${ \begin{array}{l} g(x) = (x+5)(x+4) \ h(x) = -2x + 9 \end{array} }$(a) Find { \left(\frac{g}{h}\right)(-6)$}$.(b) Find All Values That

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Suppose that the Functions g and h are Defined as Follows

Introduction

In this article, we will explore the concept of function composition and how to find the value of a composite function. We will be given two functions, g(x) and h(x), and we will need to find the value of the composite function (g/h)(x) at a specific point, x = -6. Additionally, we will need to find all values of x for which the composite function is defined.

Function Definitions

The functions g(x) and h(x) are defined as follows:

g(x)=(x+5)(x+4){ g(x) = (x+5)(x+4) } h(x)=−2x+9{ h(x) = -2x + 9 }

Part (a) - Finding the Value of the Composite Function

To find the value of the composite function (g/h)(-6), we need to follow the order of operations. First, we need to find the value of g(-6) and h(-6).

g(−6)=(−6+5)(−6+4){ g(-6) = (-6+5)(-6+4) } g(−6)=(−1)(−2){ g(-6) = (-1)(-2) } g(−6)=2{ g(-6) = 2 }

h(−6)=−2(−6)+9{ h(-6) = -2(-6) + 9 } h(−6)=12+9{ h(-6) = 12 + 9 } h(−6)=21{ h(-6) = 21 }

Now that we have the values of g(-6) and h(-6), we can find the value of the composite function (g/h)(-6).

(gh)(−6)=g(−6)h(−6){ \left(\frac{g}{h}\right)(-6) = \frac{g(-6)}{h(-6)} } (gh)(−6)=221{ \left(\frac{g}{h}\right)(-6) = \frac{2}{21} }

Part (b) - Finding All Values of x for Which the Composite Function is Defined

To find all values of x for which the composite function (g/h)(x) is defined, we need to consider the domain of the function h(x). The function h(x) is defined for all real numbers, but the function g(x) is not defined for all real numbers. The function g(x) is defined for all real numbers except for x = -5 and x = -4, because these values make the denominator of the function zero.

Therefore, the composite function (g/h)(x) is defined for all real numbers except for x = -5 and x = -4.

Conclusion

In this article, we have explored the concept of function composition and how to find the value of a composite function. We have found the value of the composite function (g/h)(-6) and all values of x for which the composite function is defined. The composite function (g/h)(x) is defined for all real numbers except for x = -5 and x = -4.

Final Answer

The final answer to part (a) is 221\boxed{\frac{2}{21}}.

The final answer to part (b) is (−∞,−5)∪(−4,∞)\boxed{(-\infty, -5) \cup (-4, \infty)}.

References

  • [1] Calculus, James Stewart, 8th edition
  • [2] Algebra, Michael Artin, 2nd edition

Related Topics

  • Function composition
  • Domain and range of a function
  • Order of operations

Keywords

  • Function composition
  • Domain and range of a function
  • Order of operations
  • Calculus
  • Algebra
    Suppose that the Functions g and h are Defined as Follows: Q&A

Introduction

In our previous article, we explored the concept of function composition and how to find the value of a composite function. We defined two functions, g(x) and h(x), and found the value of the composite function (g/h)(-6) and all values of x for which the composite function is defined. In this article, we will answer some frequently asked questions related to function composition and provide additional examples to help solidify your understanding of the concept.

Q&A

Q: What is function composition?

A: Function composition is the process of combining two or more functions to create a new function. This is done by substituting the output of one function into the input of another function.

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

A: To find the value of a composite function, you need to follow the order of operations. First, find the value of the inner function, and then substitute that value into the outer function.

Q: What is the difference between a function and a composite function?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A composite function is a new function that is created by combining two or more functions.

Q: Can I have multiple composite functions?

A: Yes, you can have multiple composite functions. For example, if you have three functions, f(x), g(x), and h(x), you can create composite functions like (f/g)(x), (g/h)(x), and (f/h)(x).

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

A: To determine the domain and range of a composite function, you need to consider the domains and ranges of the individual functions. The domain of the composite function is the set of all possible inputs that can be used to create the output, and the range is the set of all possible outputs.

Q: Can I have a composite function with multiple outputs?

A: No, a composite function can only have one output for each input. However, you can have a composite function with multiple inputs, and the output will be a single value.

Examples

Example 1:

Find the value of the composite function (f/g)(x) if f(x) = 2x + 1 and g(x) = x^2 + 1.

To find the value of the composite function, we need to follow the order of operations.

(f/g)(x)=f(x)g(x){ (f/g)(x) = \frac{f(x)}{g(x)} } (f/g)(x)=2x+1x2+1{ (f/g)(x) = \frac{2x + 1}{x^2 + 1} }

Example 2:

Find the domain and range of the composite function (f/g)(x) if f(x) = 2x + 1 and g(x) = x^2 + 1.

To determine the domain and range of the composite function, we need to consider the domains and ranges of the individual functions.

The domain of f(x) is all real numbers, and the range is all real numbers.

The domain of g(x) is all real numbers, and the range is all real numbers greater than or equal to 1.

Therefore, the domain of the composite function (f/g)(x) is all real numbers, and the range is all real numbers greater than or equal to 1.

Conclusion

In this article, we have answered some frequently asked questions related to function composition and provided additional examples to help solidify your understanding of the concept. We have also discussed how to find the value of a composite function, determine the domain and range of a composite function, and create multiple composite functions.

Final Answer

The final answer to the first question is function composition is the process of combining two or more functions to create a new function.

The final answer to the second question is to find the value of a composite function, you need to follow the order of operations.

The final answer to the third question is a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range), and a composite function is a new function that is created by combining two or more functions.

References

  • [1] Calculus, James Stewart, 8th edition
  • [2] Algebra, Michael Artin, 2nd edition

Related Topics

  • Function composition
  • Domain and range of a function
  • Order of operations

Keywords

  • Function composition
  • Domain and range of a function
  • Order of operations
  • Calculus
  • Algebra