If $h(x)=5+x$ And $k(x)=\frac{1}{x}$, Which Expression Is Equivalent To \$(k \circ H)(x)$[/tex\]?A. $\frac{(5+x)}{x}$B. $\frac{1}{(5+x)}$C. $5+\left(\frac{1}{x}\right)$D.

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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. This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis. In this article, we will explore the composition of functions, specifically the expression (k ∘ h)(x), where h(x) = 5 + x and k(x) = 1/x.

Understanding the Composition of Functions

The composition of functions is denoted by the symbol ∘. Given two functions f(x) and g(x), the composition of f and g is defined as (f ∘ g)(x) = f(g(x)). In other words, we first apply the function g to the input x, and then apply the function f to the result.

Composition of h(x) and k(x)

In this case, we are given two functions h(x) = 5 + x and k(x) = 1/x. We want to find the composition of h and k, denoted by (k ∘ h)(x). To do this, we need to apply the function h to the input x, and then apply the function k to the result.

Applying h(x) to the Input x

First, we apply the function h to the input x. This means we substitute x into the function h(x) = 5 + x, which gives us h(x) = 5 + x.

Applying k(x) to the Result

Next, we apply the function k to the result of h(x). This means we substitute h(x) into the function k(x) = 1/x. Since h(x) = 5 + x, we substitute 5 + x into the function k(x) = 1/x, which gives us k(h(x)) = 1/(5 + x).

Conclusion

Therefore, the expression (k ∘ h)(x) is equivalent to 1/(5 + x).

Answer

The correct answer is B. 1/(5 + x).

Discussion

The composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. In this case, we used the composition of functions to find the expression (k ∘ h)(x). This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis.

Real-World Applications

The composition of functions has many real-world applications, including:

  • Computer Science: The composition of functions is used in computer science to create complex algorithms and programs.
  • Engineering: The composition of functions is used in engineering to model and analyze complex systems.
  • Economics: The composition of functions is used in economics to model and analyze economic systems.

Conclusion

In conclusion, the composition of functions is a fundamental concept in mathematics that allows us to create new functions from existing ones. In this article, we explored the composition of functions, specifically the expression (k ∘ h)(x), where h(x) = 5 + x and k(x) = 1/x. We found that the expression (k ∘ h)(x) is equivalent to 1/(5 + x). This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis, and has many real-world applications.

References

  • Krantz, S. G. (2013). Calculus: An Introduction to Mathematical Analysis. Springer.
  • Rosen, K. H. (2017). Discrete Mathematics and Its Applications. McGraw-Hill.
  • Sullivan, M. (2018). Calculus: Early Transcendentals. Pearson.

Glossary

  • Composition of Functions: The composition of two functions f(x) and g(x) is denoted by (f ∘ g)(x) = f(g(x)).
  • Function: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
  • Input: The input of a function is the value that is substituted into the function.
  • Output: The output of a function is the value that is produced by the function.

Further Reading

  • Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation.
  • Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships.
  • Analysis: Analysis is a branch of mathematics that deals with the study of mathematical structures and their properties.

Introduction

In our previous article, we explored the composition of functions, specifically the expression (k ∘ h)(x), where h(x) = 5 + x and k(x) = 1/x. We found that the expression (k ∘ h)(x) is equivalent to 1/(5 + x). In this article, we will answer some frequently asked questions about the composition of functions.

Q&A

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. It is denoted by the symbol ∘.

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

A: To find the composition of two functions f(x) and g(x), you need to apply the function g to the input x, and then apply the function f to the result. This is denoted by (f ∘ g)(x) = f(g(x)).

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.

Q: Can I use the composition of functions to simplify complex expressions?

A: Yes, the composition of functions can be used to simplify complex expressions. By applying the function g to the input x, and then applying the function f to the result, you can simplify the expression and make it easier to work with.

Q: Are there any restrictions on the composition of functions?

A: Yes, there are restrictions on the composition of functions. The function g must be defined for all values of x, and the function f must be defined for all values of g(x).

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

A: Yes, the composition of functions can be used to solve equations. By applying the function g to the input x, and then applying the function f to the result, you can solve the equation and find the value of x.

Q: Are there any real-world applications of the composition of functions?

A: Yes, there are many real-world applications of the composition of functions. It is used in computer science, engineering, economics, and many other fields.

Conclusion

In conclusion, the composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. It is used in many real-world applications and can be used to simplify complex expressions and solve equations.

References

  • Krantz, S. G. (2013). Calculus: An Introduction to Mathematical Analysis. Springer.
  • Rosen, K. H. (2017). Discrete Mathematics and Its Applications. McGraw-Hill.
  • Sullivan, M. (2018). Calculus: Early Transcendentals. Pearson.

Glossary

  • Composition of Functions: The composition of two functions f(x) and g(x) is denoted by (f ∘ g)(x) = f(g(x)).
  • Function: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
  • Input: The input of a function is the value that is substituted into the function.
  • Output: The output of a function is the value that is produced by the function.

Further Reading

  • Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation.
  • Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships.
  • Analysis: Analysis is a branch of mathematics that deals with the study of mathematical structures and their properties.