In The X Y X Y X Y -plane, The Function F F F Has A Domain Of − 2 ≤ X ≤ 8 -2 \leq X \leq 8 − 2 ≤ X ≤ 8 . If The Graph Of Y = G ( X Y = G(x Y = G ( X ] Is The Result Of The Transformation G ( X ) = F ( 2 ( X + 6 ) ) − 7 G(x) = F(2(x+6)) - 7 G ( X ) = F ( 2 ( X + 6 )) − 7 , Then What Is The Domain Of G ( X G(x G ( X ]?A.

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Domain Transformation in the xyx y-plane: Understanding the Effect of g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7

When dealing with functions in the xyx y-plane, understanding the concept of domain is crucial. The domain of a function is the set of all possible input values for which the function is defined. In this article, we will explore how the domain of a function changes when it undergoes a transformation.

Understanding the Original Function ff

The original function ff has a domain of −2≤x≤8-2 \leq x \leq 8. This means that the function is defined for all values of xx between −2-2 and 88, inclusive. The graph of ff can be visualized as a curve that passes through the points (−2,f(−2))(-2, f(-2)), (−1,f(−1))(-1, f(-1)), and so on, up to (8,f(8))(8, f(8)).

The Transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7

The transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7 involves three main steps:

  1. Horizontal Shift: The expression x+6x+6 represents a horizontal shift of the graph of ff by 66 units to the left.
  2. Horizontal Stretch: The factor 22 inside the parentheses represents a horizontal stretch of the graph of ff by a factor of 22.
  3. Vertical Shift: The constant −7-7 represents a vertical shift of the graph of ff by 77 units downward.

Determining the Domain of g(x)g(x)

To determine the domain of g(x)g(x), we need to consider the effect of the transformation on the domain of ff. The horizontal shift and stretch will change the range of values for which the function is defined.

Let's analyze the transformation step by step:

  • Horizontal Shift: The expression x+6x+6 represents a horizontal shift of the graph of ff by 66 units to the left. This means that the domain of g(x)g(x) will be shifted to the left by 66 units.
  • Horizontal Stretch: The factor 22 inside the parentheses represents a horizontal stretch of the graph of ff by a factor of 22. This means that the domain of g(x)g(x) will be stretched by a factor of 22.
  • Vertical Shift: The constant −7-7 represents a vertical shift of the graph of ff by 77 units downward. This does not affect the domain of g(x)g(x).

Calculating the New Domain

To calculate the new domain of g(x)g(x), we need to apply the horizontal shift and stretch to the original domain of ff.

The original domain of ff is −2≤x≤8-2 \leq x \leq 8. After applying the horizontal shift of 66 units to the left, the new domain becomes −2−6≤x≤8−6-2 - 6 \leq x \leq 8 - 6, which simplifies to −8≤x≤2-8 \leq x \leq 2.

Next, we apply the horizontal stretch by a factor of 22. This means that the new domain will be stretched by a factor of 22, resulting in −8/2≤x≤2/2-8/2 \leq x \leq 2/2, which simplifies to −4≤x≤1-4 \leq x \leq 1.

Therefore, the domain of g(x)g(x) is −4≤x≤1-4 \leq x \leq 1.

Conclusion

In conclusion, the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7 affects the domain of the original function ff in a predictable way. By applying the horizontal shift and stretch, we can calculate the new domain of g(x)g(x). In this case, the domain of g(x)g(x) is −4≤x≤1-4 \leq x \leq 1. Understanding the effect of transformations on the domain of a function is essential in mathematics and has numerous applications in various fields.

Key Takeaways

  • The domain of a function is the set of all possible input values for which the function is defined.
  • The transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7 involves a horizontal shift, horizontal stretch, and vertical shift.
  • The horizontal shift and stretch affect the domain of the function, while the vertical shift does not.
  • To calculate the new domain of g(x)g(x), we need to apply the horizontal shift and stretch to the original domain of ff.

Further Reading

For more information on transformations and their effects on the domain of a function, we recommend exploring the following topics:

  • Horizontal shifts and stretches
  • Vertical shifts
  • Domain and range of functions
  • Graphing functions in the xyx y-plane

By understanding the concept of domain transformation, you can better analyze and solve problems involving functions in mathematics and other fields.
Domain Transformation Q&A: Understanding the Effect of g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7

In our previous article, we explored the concept of domain transformation and how it affects the domain of a function. We analyzed the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7 and determined that the domain of g(x)g(x) is −4≤x≤1-4 \leq x \leq 1. In this article, we will answer some frequently asked questions about domain transformation and provide additional insights.

Q: What is the difference between a horizontal shift and a horizontal stretch?

A: A horizontal shift is a transformation that moves the graph of a function to the left or right by a certain number of units. In the case of the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7, the horizontal shift is 66 units to the left. A horizontal stretch, on the other hand, is a transformation that stretches the graph of a function horizontally by a certain factor. In this case, the horizontal stretch is by a factor of 22.

Q: How do I determine the new domain of a function after a transformation?

A: To determine the new domain of a function after a transformation, you need to apply the transformation to the original domain of the function. In the case of the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7, we applied the horizontal shift and stretch to the original domain of ff to determine the new domain of g(x)g(x).

Q: What is the effect of a vertical shift on the domain of a function?

A: A vertical shift does not affect the domain of a function. It only changes the position of the graph of the function vertically. In the case of the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7, the vertical shift is −7-7 units downward, but it does not affect the domain of g(x)g(x).

Q: Can I apply multiple transformations to a function?

A: Yes, you can apply multiple transformations to a function. However, you need to apply them in the correct order. In the case of the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7, we applied the horizontal shift and stretch first, and then the vertical shift.

Q: How do I graph a function after a transformation?

A: To graph a function after a transformation, you need to apply the transformation to the original graph of the function. In the case of the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7, you would graph the original function ff and then apply the horizontal shift and stretch to get the graph of g(x)g(x).

Q: What are some common transformations that affect the domain of a function?

A: Some common transformations that affect the domain of a function include:

  • Horizontal shifts: moving the graph of a function to the left or right by a certain number of units
  • Horizontal stretches: stretching the graph of a function horizontally by a certain factor
  • Vertical shifts: moving the graph of a function vertically by a certain number of units
  • Reflections: reflecting the graph of a function across the x-axis or y-axis

Q: How do I determine the range of a function after a transformation?

A: To determine the range of a function after a transformation, you need to apply the transformation to the original range of the function. In the case of the transformation g(x)=f(2(x+6))−7g(x) = f(2(x+6)) - 7, we would apply the transformation to the original range of ff to determine the range of g(x)g(x).

Conclusion

In conclusion, domain transformation is an essential concept in mathematics that affects the domain of a function. By understanding the effect of transformations on the domain of a function, you can better analyze and solve problems involving functions in mathematics and other fields. We hope that this Q&A article has provided you with a better understanding of domain transformation and its applications.

Key Takeaways

  • Domain transformation affects the domain of a function.
  • Horizontal shifts and stretches affect the domain of a function.
  • Vertical shifts do not affect the domain of a function.
  • Multiple transformations can be applied to a function.
  • Graphing a function after a transformation involves applying the transformation to the original graph of the function.

Further Reading

For more information on domain transformation and its applications, we recommend exploring the following topics:

  • Horizontal shifts and stretches
  • Vertical shifts
  • Domain and range of functions
  • Graphing functions in the xyx y-plane
  • Transformations and their effects on the domain of a function