Select All The Correct Answers.The Function F ( X ) = X 1 3 F(x)=x^{\frac{1}{3}} F ( X ) = X 3 1 ​ Is Transformed To Get Function H H H , Where H ( X ) = ( 2 X ) 1 T + 5 H(x)=(2x)^{\frac{1}{t}}+5 H ( X ) = ( 2 X ) T 1 ​ + 5 .Which Statements Are True About Function H H H ?A. As X X X Approaches

In mathematics, function transformations are essential concepts that help us understand how functions change under various operations. In this article, we will explore the transformation of the function $f(x)=x^$\frac{1}{3}}x31​ to get function $h$, where $h(x)=(2x)^{\frac{1{3}}+5$. We will examine the properties of function $h$ and determine which statements are true about it.

Function $f(x)$ and its Transformation

The original function is $f(x)=x^{\frac{1}{3}}$. To transform this function into $h(x)$, we need to apply two operations: a horizontal stretch by a factor of $\frac{1}{2}$ and a vertical shift upward by 5 units.

Horizontal Stretch

The horizontal stretch is achieved by replacing $x$ with $2x$ in the original function. This means that the input $x$ is multiplied by 2, resulting in a horizontal stretch by a factor of $\frac{1}{2}$.

Vertical Shift

The vertical shift is achieved by adding 5 to the transformed function. This means that the output of the function is shifted upward by 5 units.

Function $h(x)$

After applying the horizontal stretch and vertical shift, we get the transformed function $h(x)=(2x)^{\frac{1}{3}}+5$.

Properties of Function $h(x)$

Now that we have the transformed function $h(x)$, let's examine its properties.

Domain of Function $h(x)$

The domain of a function is the set of all possible input values for which the function is defined. In the case of function $h(x)$, the input value $x$ must be non-negative, since the cube root of a negative number is undefined.

Range of Function $h(x)$

The range of a function is the set of all possible output values for which the function is defined. In the case of function $h(x)$, the output value is always greater than or equal to 5, since the cube root of any non-negative number is greater than or equal to 0, and adding 5 shifts the output upward.

End Behavior of Function $h(x)$

The end behavior of a function refers to the behavior of the function as the input value approaches positive or negative infinity. In the case of function $h(x)$, as $x$ approaches positive infinity, the cube root of $2x$ approaches infinity, and adding 5 shifts the output upward. As $x$ approaches negative infinity, the cube root of $2x$ approaches negative infinity, and adding 5 shifts the output upward.

Asymptotes of Function $h(x)$

An asymptote is a line that the graph of a function approaches as the input value approaches positive or negative infinity. In the case of function $h(x)$, there is no horizontal asymptote, since the cube root of $2x$ approaches infinity as $x$ approaches positive or negative infinity. However, there is a vertical asymptote at $x=0$, since the cube root of $2x$ is undefined when $x=0$.

Increasing or Decreasing Nature of Function $h(x)$

The increasing or decreasing nature of a function refers to whether the function is increasing or decreasing over a given interval. In the case of function $h(x)$, the function is increasing over the interval $[0,\infty)$, since the cube root of $2x$ is an increasing function over this interval.

Statements about Function $h(x)$

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Now that we have examined the properties of function $h(x)$, let's determine which statements are true about it.

A. As $x$ approaches positive infinity, $h(x)$ approaches infinity.

This statement is true, since the cube root of $2x$ approaches infinity as $x$ approaches positive infinity, and adding 5 shifts the output upward.

B. As $x$ approaches negative infinity, $h(x)$ approaches negative infinity.

This statement is false, since the cube root of $2x$ approaches negative infinity as $x$ approaches negative infinity, but adding 5 shifts the output upward, resulting in a positive value.

C. The domain of function $h(x)$ is all real numbers.

This statement is false, since the input value $x$ must be non-negative, since the cube root of a negative number is undefined.

D. The range of function $h(x)$ is all real numbers.

This statement is false, since the output value is always greater than or equal to 5, since the cube root of any non-negative number is greater than or equal to 0, and adding 5 shifts the output upward.

E. The end behavior of function $h(x)$ is the same as the end behavior of function $f(x)$.

This statement is false, since the end behavior of function $h(x)$ is different from the end behavior of function $f(x)$, since the horizontal stretch and vertical shift operations change the end behavior of the function.

F. The asymptotes of function $h(x)$ are the same as the asymptotes of function $f(x)$.

This statement is false, since the asymptotes of function $h(x)$ are different from the asymptotes of function $f(x)$, since the horizontal stretch and vertical shift operations change the asymptotes of the function.

G. The increasing or decreasing nature of function $h(x)$ is the same as the increasing or decreasing nature of function $f(x)$.

This statement is false, since the increasing or decreasing nature of function $h(x)$ is different from the increasing or decreasing nature of function $f(x)$, since the horizontal stretch and vertical shift operations change the increasing or decreasing nature of the function.

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In our previous article, we explored the transformation of the function $f(x)=x^{\frac{1}{3}}$ to get function $h$, where $h(x)=(2x)^{\frac{1}{3}}+5$. We examined the properties of function $h$ and determined which statements are true about it. In this article, we will answer some frequently asked questions about function transformations in mathematics.

Q: What is a function transformation?

A function transformation is a change in the function that results in a new function. This can be achieved by applying various operations such as horizontal and vertical shifts, stretches, and compressions.

Q: What are the different types of function transformations?

There are several types of function transformations, including:

• Horizontal shifts: Moving the function to the left or right by a certain amount.
• Vertical shifts: Moving the function up or down by a certain amount.
• Horizontal stretches: Stretching the function horizontally by a certain factor.
• Horizontal compressions: Compressing the function horizontally by a certain factor.
• Vertical stretches: Stretching the function vertically by a certain factor.
• Vertical compressions: Compressing the function vertically by a certain factor.

Q: How do I determine the type of function transformation?

To determine the type of function transformation, you need to examine the function and identify the changes that have been made. For example, if the function has been shifted to the left by 2 units, you can determine that it is a horizontal shift.

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

A horizontal stretch is a transformation that stretches the function horizontally by a certain factor, while a horizontal compression is a transformation that compresses the function horizontally by a certain factor.

Q: How do I apply a horizontal stretch or compression to a function?

To apply a horizontal stretch or compression to a function, you need to multiply the input value by a certain factor. For example, if you want to apply a horizontal stretch by a factor of 2, you would multiply the input value by 2.

Q: What is the difference between a vertical stretch and a vertical compression?

A vertical stretch is a transformation that stretches the function vertically by a certain factor, while a vertical compression is a transformation that compresses the function vertically by a certain factor.

Q: How do I apply a vertical stretch or compression to a function?

To apply a vertical stretch or compression to a function, you need to multiply the output value by a certain factor. For example, if you want to apply a vertical stretch by a factor of 2, you would multiply the output value by 2.

Q: Can I apply multiple function transformations to a function?

Yes, you can apply multiple function transformations to a function. For example, you can apply a horizontal shift and a vertical stretch to a function.

Q: How do I determine the order of function transformations?

To determine the order of function transformations, you need to examine the function and identify the transformations that have been applied. For example, if a function has been shifted to the left by 2 units and then stretched vertically by a factor of 2, you can determine that the horizontal shift was applied first and the vertical stretch was applied second.

Q: Can I undo a function transformation?

Yes, you can undo a function transformation by applying the inverse transformation. For example, if a function has been shifted to the left by 2 units, you can undo the shift by shifting the function to the right by 2 units.

Q: What are some common function transformations?

Some common function transformations include:

• Reflection: Reflecting the function across a certain line or axis.
• Rotation: Rotating the function by a certain angle.
• Scaling: Scaling the function by a certain factor.
• Translation: Translating the function by a certain amount.

Q: How do I apply a reflection to a function?

To apply a reflection to a function, you need to reflect the function across a certain line or axis. For example, if you want to reflect a function across the x-axis, you would multiply the output value by -1.

Q: How do I apply a rotation to a function?

To apply a rotation to a function, you need to rotate the function by a certain angle. For example, if you want to rotate a function by 90 degrees counterclockwise, you would multiply the input value by -1 and the output value by -1.

Q: How do I apply a scaling to a function?

To apply a scaling to a function, you need to scale the function by a certain factor. For example, if you want to scale a function by a factor of 2, you would multiply the input value by 2 and the output value by 2.

Q: How do I apply a translation to a function?

To apply a translation to a function, you need to translate the function by a certain amount. For example, if you want to translate a function to the left by 2 units, you would subtract 2 from the input value.

In conclusion, we have answered some frequently asked questions about function transformations in mathematics. We have explored the different types of function transformations, how to apply them, and how to determine the order of function transformations. We have also discussed some common function transformations and how to apply them.