Given The Functions:$ \begin{array}{l} f(x) = 2x^2 - X + 1 \ g(x) = 4x + 3 \end{array} }$a. Find { F \circ G $}$ And { G \circ F $}$.(i) Instructions - { F \circ G $ $ Represents The

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Composition of Functions: Finding f โˆ˜ g and g โˆ˜ f

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. Given two functions, f(x) and g(x), the composition of f and g, denoted as f โˆ˜ g, is defined as f โˆ˜ g = f(g(x)). Similarly, the composition of g and f, denoted as g โˆ˜ f, is defined as g โˆ˜ f = g(f(x)). In this article, we will explore the composition of two given functions, f(x) = 2x^2 - x + 1 and g(x) = 4x + 3, and find the expressions for f โˆ˜ g and g โˆ˜ f.

Step 1: Finding f โˆ˜ g

To find f โˆ˜ g, we need to substitute g(x) into f(x) in place of x. This means we will replace every instance of x in f(x) with g(x).

f(x)=2x2โˆ’x+1{ f(x) = 2x^2 - x + 1 }

Substituting g(x) into f(x):

[fโˆ˜g](x)=2(g(x))2โˆ’g(x)+1{ [f โˆ˜ g](x) = 2(g(x))^2 - g(x) + 1 }

Now, we will substitute the expression for g(x) into the equation:

g(x)=4x+3{ g(x) = 4x + 3 }

Substituting g(x) into f โˆ˜ g:

[fโˆ˜g](x)=2(4x+3)2โˆ’(4x+3)+1{ [f โˆ˜ g](x) = 2(4x + 3)^2 - (4x + 3) + 1 }

Expanding the squared term:

[fโˆ˜g](x)=2(16x2+24x+9)โˆ’4xโˆ’3+1{ [f โˆ˜ g](x) = 2(16x^2 + 24x + 9) - 4x - 3 + 1 }

Distributing the 2:

[fโˆ˜g](x)=32x2+48x+18โˆ’4xโˆ’3+1{ [f โˆ˜ g](x) = 32x^2 + 48x + 18 - 4x - 3 + 1 }

Combining like terms:

[fโˆ˜g](x)=32x2+44x+16{ [f โˆ˜ g](x) = 32x^2 + 44x + 16 }

Step 2: Finding g โˆ˜ f

To find g โˆ˜ f, we need to substitute f(x) into g(x) in place of x. This means we will replace every instance of x in g(x) with f(x).

g(x)=4x+3{ g(x) = 4x + 3 }

Substituting f(x) into g(x):

[gโˆ˜f](x)=4(f(x))+3{ [g โˆ˜ f](x) = 4(f(x)) + 3 }

Now, we will substitute the expression for f(x) into the equation:

f(x)=2x2โˆ’x+1{ f(x) = 2x^2 - x + 1 }

Substituting f(x) into g โˆ˜ f:

[gโˆ˜f](x)=4(2x2โˆ’x+1)+3{ [g โˆ˜ f](x) = 4(2x^2 - x + 1) + 3 }

Distributing the 4:

[gโˆ˜f](x)=8x2โˆ’4x+4+3{ [g โˆ˜ f](x) = 8x^2 - 4x + 4 + 3 }

Combining like terms:

[gโˆ˜f](x)=8x2โˆ’4x+7{ [g โˆ˜ f](x) = 8x^2 - 4x + 7 }

Conclusion

In this article, we have found the expressions for f โˆ˜ g and g โˆ˜ f given the functions f(x) = 2x^2 - x + 1 and g(x) = 4x + 3. The expression for f โˆ˜ g is 32x^2 + 44x + 16, and the expression for g โˆ˜ f is 8x^2 - 4x + 7. These expressions demonstrate the concept of composition of functions and how it can be used to create new functions from existing ones.

Key Takeaways

  • The composition of functions is a fundamental concept in mathematics that allows us to combine two or more functions to create a new function.
  • The composition of f and g, denoted as f โˆ˜ g, is defined as f โˆ˜ g = f(g(x)).
  • The composition of g and f, denoted as g โˆ˜ f, is defined as g โˆ˜ f = g(f(x)).
  • To find f โˆ˜ g, we need to substitute g(x) into f(x) in place of x.
  • To find g โˆ˜ f, we need to substitute f(x) into g(x) in place of x.

Further Reading

  • Composition of functions is a fundamental concept in mathematics and is used extensively in various fields such as physics, engineering, and computer science.
  • The composition of functions can be used to model real-world phenomena and can be used to solve complex problems.
  • The concept of composition of functions can be extended to more than two functions, and can be used to create complex functions from simpler ones.
    Composition of Functions: Q&A

Introduction

In our previous article, we explored the concept of composition of functions and found the expressions for f โˆ˜ g and g โˆ˜ f given the functions f(x) = 2x^2 - x + 1 and g(x) = 4x + 3. In this article, we will answer some frequently asked questions about composition of functions and provide additional insights into this important concept.

Q: What is the difference between f โˆ˜ g and g โˆ˜ f?

A: The main difference between f โˆ˜ g and g โˆ˜ f is the order in which the functions are composed. In f โˆ˜ g, we first apply the function g(x) and then apply the function f(x) to the result. In g โˆ˜ f, we first apply the function f(x) and then apply the function g(x) to the result.

Q: How do I know which function to compose first?

A: The order in which you compose the functions depends on the specific problem you are trying to solve. In general, if you are given two functions f(x) and g(x), you can try composing them in both orders and see which one gives you the desired result.

Q: Can I compose more than two functions?

A: Yes, you can compose more than two functions. For example, if you have three functions f(x), g(x), and h(x), you can compose them in the following order: f โˆ˜ g โˆ˜ h = f(g(h(x))). This is known as a chain of composition.

Q: What is the domain and range of a composite function?

A: The domain of a composite function f โˆ˜ g is the set of all values of x for which g(x) is defined. The range of f โˆ˜ g is the set of all values of f(g(x)).

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

A: Yes, composition of functions can be used to solve equations. For example, if you have an equation of the form f(x) = g(x), you can try composing the functions f(x) and g(x) to see if you can simplify the equation.

Q: Are there any special properties of composite functions?

A: Yes, there are several special properties of composite functions. For example, if f(x) and g(x) are both one-to-one functions, then f โˆ˜ g is also one-to-one. Additionally, if f(x) and g(x) are both onto functions, then f โˆ˜ g is also onto.

Q: Can I use composition of functions to model real-world phenomena?

A: Yes, composition of functions can be used to model real-world phenomena. For example, if you are studying the motion of an object, you can use composition of functions to model the position, velocity, and acceleration of the object over time.

Q: Are there any limitations to composition of functions?

A: Yes, there are several limitations to composition of functions. For example, if you are composing functions that are not defined for all values of x, you may need to restrict the domain of the composite function. Additionally, if you are composing functions that are not invertible, you may not be able to find a unique solution to the equation.

Conclusion

In this article, we have answered some frequently asked questions about composition of functions and provided additional insights into this important concept. We have also discussed some of the special properties of composite functions and how they can be used to model real-world phenomena. By understanding composition of functions, you can solve a wide range of problems in mathematics, science, and engineering.

Key Takeaways

  • Composition of functions is a powerful tool for solving equations and modeling real-world phenomena.
  • The order in which you compose functions depends on the specific problem you are trying to solve.
  • You can compose more than two functions to create a chain of composition.
  • The domain and range of a composite function depend on the domain and range of the individual functions.
  • Composition of functions can be used to solve equations and model real-world phenomena.

Further Reading

  • Composition of functions is a fundamental concept in mathematics and is used extensively in various fields such as physics, engineering, and computer science.
  • The composition of functions can be used to model complex systems and solve complex problems.
  • The concept of composition of functions can be extended to more than two functions, and can be used to create complex functions from simpler ones.