The Function $f(x) = X^{\frac{1}{2}}$ Is Transformed To Get Function $w(x) = -(3x)^{\frac{1}{2}} - 4$.What Are The Domain And The Range Of Function \$w(x)$[/tex\]?Domain: $x \geq$ $ \square$Range:

Introduction

In mathematics, function transformation is a crucial concept that helps us understand how a function changes under various operations. In this article, we will explore the transformation of the function $f(x) = x^{\frac{1}{2}}$ to get the function $w(x) = -(3x)^{\frac{1}{2}} - 4$. We will analyze the domain and range of the transformed function $w(x)$.

Understanding the Original Function

The original function is $f(x) = x^{\frac{1}{2}}$. This is a square root function, which means it takes the square root of the input value $x$. The domain of this function is all non-negative real numbers, i.e., $x \geq 0$. The range of this function is all non-negative real numbers, i.e., $y \geq 0$.

Transformation of the Function

The transformed function is $w(x) = -(3x)^{\frac{1}{2}} - 4$. To understand this transformation, let's break it down:

• The function $f(x)$ is multiplied by $3$, resulting in $3x$.
• The square root of $3x$ is taken, resulting in $(3x)^{\frac{1}{2}}$.
• The negative sign is applied to the result, resulting in $-(3x)^{\frac{1}{2}}$.
• Finally, $4$ is subtracted from the result, resulting in $-(3x)^{\frac{1}{2}} - 4$.

Domain Analysis

To find the domain of the transformed function $w(x)$, we need to consider the restrictions imposed by the square root and the negative sign.

• The square root of $3x$ is defined only when $3x \geq 0$. This implies that $x \geq 0$.
• The negative sign does not impose any additional restrictions on the domain.

Therefore, the domain of the transformed function $w(x)$ is $x \geq 0$.

Range Analysis

To find the range of the transformed function $w(x)$, we need to consider the effect of the negative sign and the constant term $-4$.

• The negative sign flips the range of the original function $f(x)$, resulting in a range of $y \leq 0$.
• The constant term $-4$ shifts the range of the original function $f(x)$ down by $4$ units, resulting in a range of $y \leq -4$.

Therefore, the range of the transformed function $w(x)$ is $y \leq -4$.

Conclusion

In conclusion, the transformed function $w(x) = -(3x)^{\frac{1}{2}} - 4$ has a domain of $x \geq 0$ and a range of $y \leq -4$. The transformation of the original function $f(x) = x^{\frac{1}{2}}$ involves multiplying by $3$, taking the square root, applying a negative sign, and subtracting $4$.

Domain and Range Summary

Final Thoughts

Introduction

In our previous article, we explored the transformation of the function $f(x) = x^{\frac{1}{2}}$ to get the function $w(x) = -(3x)^{\frac{1}{2}} - 4$. We analyzed the domain and range of the transformed function $w(x)$. In this article, we will answer some frequently asked questions related to function transformation and domain/range analysis.

Q1: What is function transformation?

A: Function transformation is a process of changing a function into a new function by applying various operations such as multiplication, division, addition, subtraction, and exponentiation.

Q2: How do you determine the domain of a transformed function?

A: To determine the domain of a transformed function, you need to consider the restrictions imposed by the operations applied to the original function. For example, if the original function has a domain of $x \geq 0$ and the transformed function involves taking the square root of $x$, the domain of the transformed function will be $x \geq 0$.

Q3: How do you determine the range of a transformed function?

A: To determine the range of a transformed function, you need to consider the effect of the operations applied to the original function. For example, if the original function has a range of $y \geq 0$ and the transformed function involves multiplying by a negative number, the range of the transformed function will be $y \leq 0$.

Q4: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values.

Q5: How do you graph a transformed function?

A: To graph a transformed function, you need to apply the same transformations to the graph of the original function. For example, if the original function is a square root function and the transformed function involves multiplying by $3$, the graph of the transformed function will be a square root function with a horizontal stretch by a factor of $3$.

Q6: Can you give an example of a transformed function with a different domain and range?

A: Yes, consider the function $f(x) = x^2$. The domain of this function is all real numbers, i.e., $x \in \mathbb{R}$, and the range is all non-negative real numbers, i.e., $y \geq 0$. Now, consider the transformed function $w(x) = -(x-2)^2 + 1$. The domain of this function is all real numbers, i.e., $x \in \mathbb{R}$, but the range is all real numbers less than or equal to $1$, i.e., $y \leq 1$.

Q7: How do you determine the vertex of a transformed function?

A: To determine the vertex of a transformed function, you need to apply the same transformations to the vertex of the original function. For example, if the original function is a quadratic function with a vertex at $(h, k)$ and the transformed function involves a horizontal stretch by a factor of $3$, the vertex of the transformed function will be at $(3h, k)$.

Q8: Can you give an example of a transformed function with a different vertex?

A: Yes, consider the function $f(x) = (x-2)^2$. The vertex of this function is at $(2, 0)$. Now, consider the transformed function $w(x) = -(x-4)^2 + 1$. The vertex of this function is at $(4, 1)$.

Conclusion

In conclusion, function transformation and domain/range analysis are essential concepts in mathematics. By understanding how to determine the domain and range of a transformed function, you can apply this knowledge to a wide range of mathematical problems. We hope this Q&A article has provided you with a better understanding of these concepts.

Domain and Range Summary

Final Thoughts

In this article, we answered some frequently asked questions related to function transformation and domain/range analysis. We hope this article has provided you with a better understanding of these concepts and has helped you to apply this knowledge to your own mathematical problems.