Calculating the New Domain

Oct 13, 2025 by miodeya.com 18 views

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

When dealing with functions in the $x y$ -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 $f$

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

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

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

Horizontal Shift: The expression $x+6$ represents a horizontal shift of the graph of $f$ by $6$ units to the left.
Horizontal Stretch: The factor $2$ inside the parentheses represents a horizontal stretch of the graph of $f$ by a factor of $2$ .
Vertical Shift: The constant $-7$ represents a vertical shift of the graph of $f$ by $7$ units downward.

Determining the Domain of $g(x)$

To determine the domain of $g(x)$ , we need to consider the effect of the transformation on the domain of $f$ . 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 $x+6$ represents a horizontal shift of the graph of $f$ by $6$ units to the left. This means that the domain of $g(x)$ will be shifted to the left by $6$ units.
Horizontal Stretch: The factor $2$ inside the parentheses represents a horizontal stretch of the graph of $f$ by a factor of $2$ . This means that the domain of $g(x)$ will be stretched by a factor of $2$ .
Vertical Shift: The constant $-7$ represents a vertical shift of the graph of $f$ by $7$ units downward. This does not affect the domain of $g(x)$ .

Calculating the New Domain

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

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

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

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

Conclusion

In conclusion, the transformation $g(x) = f(2(x+6)) - 7$ affects the domain of the original function $f$ in a predictable way. By applying the horizontal shift and stretch, we can calculate the new domain of $g(x)$ . In this case, the domain of $g(x)$ is $-4 \leq x \leq 1$ . 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 $g(x) = f(2(x+6)) - 7$ 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 $g(x)$ , we need to apply the horizontal shift and stretch to the original domain of $f$ .

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 $x y$ -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 $g(x) = f(2(x+6)) - 7$

In our previous article, we explored the concept of domain transformation and how it affects the domain of a function. We analyzed the transformation $g(x) = f(2(x+6)) - 7$ and determined that the domain of $g(x)$ is $-4 \leq x \leq 1$ . 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 $g(x) = f(2(x+6)) - 7$ , the horizontal shift is $6$ 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 $2$ .

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 $g(x) = f(2(x+6)) - 7$ , we applied the horizontal shift and stretch to the original domain of $f$ to determine the new domain of $g(x)$ .

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 $g(x) = f(2(x+6)) - 7$ , the vertical shift is $-7$ units downward, but it does not affect the domain of $g(x)$ .

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 $g(x) = f(2(x+6)) - 7$ , 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 $g(x) = f(2(x+6)) - 7$ , you would graph the original function $f$ and then apply the horizontal shift and stretch to get the graph of $g(x)$ .

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 $g(x) = f(2(x+6)) - 7$ , we would apply the transformation to the original range of $f$ to determine the range of $g(x)$ .

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 $x y$ -plane
Transformations and their effects on the domain of a function