Find The Composition Of The Function Given The Following:Functions:- $f(x) = X^{\frac{1}{2}}$- $g(x) = \frac{1}{x}$Find $g(f(x)$\].Options:A. $x^2$B. $x^{\frac{1}{2}}$C. $x^{-\frac{1}{2}}$D.

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**Composition of Functions: A Step-by-Step Guide** =====================================================

What is Composition of Functions?

Composition of functions is a process of combining two or more functions to create a new function. This new function takes the output of one function as the input for another function. In other words, it is a way of combining functions to create a new function that can be used to solve problems.

Why is Composition of Functions Important?

Composition of functions is an important concept in mathematics and computer science. It is used to solve complex problems by breaking them down into smaller, more manageable parts. It is also used to create new functions that can be used to solve problems in a variety of fields, including physics, engineering, and economics.

How to Find the Composition of Functions

To find the composition of two functions, we need to follow these steps:

  1. Identify the two functions: We need to identify the two functions that we want to compose. In this case, we have two functions: f(x)=x12f(x) = x^{\frac{1}{2}} and g(x)=1xg(x) = \frac{1}{x}.
  2. Substitute the output of one function into the other: We need to substitute the output of one function into the other function. In this case, we will substitute the output of f(x)f(x) into g(x)g(x).
  3. Simplify the expression: We need to simplify the expression to get the final answer.

Example: Finding the Composition of f(x)f(x) and g(x)g(x)

Let's find the composition of f(x)f(x) and g(x)g(x).

f(x)=x12f(x) = x^{\frac{1}{2}}

g(x)=1xg(x) = \frac{1}{x}

To find the composition of f(x)f(x) and g(x)g(x), we need to substitute the output of f(x)f(x) into g(x)g(x).

g(f(x))=1f(x)g(f(x)) = \frac{1}{f(x)}

g(f(x))=1x12g(f(x)) = \frac{1}{x^{\frac{1}{2}}}

g(f(x))=x−12g(f(x)) = x^{-\frac{1}{2}}

Therefore, the composition of f(x)f(x) and g(x)g(x) is x−12x^{-\frac{1}{2}}.

Q&A

Q: What is the composition of f(x)=x2f(x) = x^2 and g(x)=1xg(x) = \frac{1}{x}? A: To find the composition of f(x)f(x) and g(x)g(x), we need to substitute the output of f(x)f(x) into g(x)g(x). g(f(x))=1f(x)=1x2=x−2g(f(x)) = \frac{1}{f(x)} = \frac{1}{x^2} = x^{-2}.

Q: What is the composition of f(x)=1xf(x) = \frac{1}{x} and g(x)=x2g(x) = x^2? A: To find the composition of f(x)f(x) and g(x)g(x), we need to substitute the output of f(x)f(x) into g(x)g(x). g(f(x))=f(x)2=(1x)2=1x2g(f(x)) = f(x)^2 = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2}.

Q: What is the composition of f(x)=x3f(x) = x^3 and g(x)=1xg(x) = \frac{1}{x}? A: To find the composition of f(x)f(x) and g(x)g(x), we need to substitute the output of f(x)f(x) into g(x)g(x). g(f(x))=1f(x)=1x3=x−3g(f(x)) = \frac{1}{f(x)} = \frac{1}{x^3} = x^{-3}.

Q: What is the composition of f(x)=1xf(x) = \frac{1}{x} and g(x)=x3g(x) = x^3? A: To find the composition of f(x)f(x) and g(x)g(x), we need to substitute the output of f(x)f(x) into g(x)g(x). g(f(x))=f(x)3=(1x)3=1x3g(f(x)) = f(x)^3 = \left(\frac{1}{x}\right)^3 = \frac{1}{x^3}.

Conclusion

In conclusion, composition of functions is an important concept in mathematics and computer science. It is used to solve complex problems by breaking them down into smaller, more manageable parts. By following the steps outlined in this article, we can find the composition of two functions and use it to solve problems in a variety of fields.

Common Mistakes to Avoid

  • Not following the order of operations: When finding the composition of two functions, we need to follow the order of operations. This means that we need to substitute the output of one function into the other function before simplifying the expression.
  • Not simplifying the expression: When finding the composition of two functions, we need to simplify the expression to get the final answer. This means that we need to combine like terms and eliminate any unnecessary variables.

Tips and Tricks

  • Use a calculator: When finding the composition of two functions, we can use a calculator to simplify the expression and get the final answer.
  • Check your work: When finding the composition of two functions, we need to check our work to make sure that we have followed the correct steps and obtained the correct answer.

Practice Problems

  • Find the composition of f(x)=x2f(x) = x^2 and g(x)=1xg(x) = \frac{1}{x}.
  • Find the composition of f(x)=1xf(x) = \frac{1}{x} and g(x)=x2g(x) = x^2.
  • Find the composition of f(x)=x3f(x) = x^3 and g(x)=1xg(x) = \frac{1}{x}.
  • Find the composition of f(x)=1xf(x) = \frac{1}{x} and g(x)=x3g(x) = x^3.

Answer Key

  • Find the composition of f(x)=x2f(x) = x^2 and g(x)=1xg(x) = \frac{1}{x}: x−2x^{-2}
  • Find the composition of f(x)=1xf(x) = \frac{1}{x} and g(x)=x2g(x) = x^2: 1x2\frac{1}{x^2}
  • Find the composition of f(x)=x3f(x) = x^3 and g(x)=1xg(x) = \frac{1}{x}: x−3x^{-3}
  • Find the composition of f(x)=1xf(x) = \frac{1}{x} and g(x)=x3g(x) = x^3: 1x3\frac{1}{x^3}