Which Best Describes The Range Of The Function F ( X ) = 2 3 ( 6 ) X F(x) = \frac{2}{3}(6)^x F ( X ) = 3 2 ​ ( 6 ) X After It Has Been Reflected Over The X X X -axis?A. All Real Numbers B. All Real Numbers Less Than 0 C. All Real Numbers Greater Than 0 D. All Real Numbers Less

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Introduction

In mathematics, reflecting a function over the x-axis is a fundamental concept that helps us understand the behavior of functions and their transformations. When a function is reflected over the x-axis, its graph is flipped upside down, and the y-values are negated. In this article, we will explore the concept of reflecting a function over the x-axis and apply it to the given function f(x)=23(6)xf(x) = \frac{2}{3}(6)^x.

What is Reflection over the x-axis?

Reflection over the x-axis is a transformation that flips a function's graph upside down. When a function is reflected over the x-axis, its y-values are negated, and the graph is flipped. This means that if a point (x, y) is on the original graph, the corresponding point on the reflected graph is (x, -y).

How to Reflect a Function over the x-axis?

To reflect a function over the x-axis, we can use the following steps:

  1. Negate the y-values of the function.
  2. Flip the graph upside down.

Reflection of the Function f(x)=23(6)xf(x) = \frac{2}{3}(6)^x

Now, let's apply the concept of reflection over the x-axis to the given function f(x)=23(6)xf(x) = \frac{2}{3}(6)^x. To reflect this function over the x-axis, we need to negate the y-values of the function.

Step 1: Negate the y-values of the function

The original function is f(x)=23(6)xf(x) = \frac{2}{3}(6)^x. To negate the y-values, we multiply the function by -1.

−f(x)=−23(6)x-f(x) = -\frac{2}{3}(6)^x

Step 2: Flip the graph upside down

Since we have negated the y-values, the graph is now flipped upside down. This means that if a point (x, y) is on the original graph, the corresponding point on the reflected graph is (x, -y).

Range of the Reflected Function

Now that we have reflected the function over the x-axis, we need to find the range of the reflected function. The range of a function is the set of all possible y-values that the function can take.

Analyzing the Reflected Function

Let's analyze the reflected function −f(x)=−23(6)x-f(x) = -\frac{2}{3}(6)^x. Since the function is an exponential function, it has a base of 6, which is greater than 1. This means that the function will increase exponentially as x increases.

Finding the Range of the Reflected Function

To find the range of the reflected function, we need to consider the behavior of the function as x approaches positive and negative infinity.

As x approaches positive infinity

As x approaches positive infinity, the function −f(x)=−23(6)x-f(x) = -\frac{2}{3}(6)^x will approach negative infinity. This is because the exponential function (6)x(6)^x will grow without bound, and the negative sign will make the function approach negative infinity.

As x approaches negative infinity

As x approaches negative infinity, the function −f(x)=−23(6)x-f(x) = -\frac{2}{3}(6)^x will approach positive infinity. This is because the exponential function (6)x(6)^x will grow without bound, and the negative sign will make the function approach positive infinity.

Conclusion

In conclusion, the range of the reflected function −f(x)=−23(6)x-f(x) = -\frac{2}{3}(6)^x is all real numbers. This is because the function will approach negative infinity as x approaches positive infinity and approach positive infinity as x approaches negative infinity.

Answer

The correct answer is A. all real numbers.

Final Thoughts

Introduction

In our previous article, we explored the concept of reflecting a function over the x-axis and applied it to the given function f(x)=23(6)xf(x) = \frac{2}{3}(6)^x. In this article, we will answer some frequently asked questions about reflecting functions over the x-axis.

Q: What is the purpose of reflecting a function over the x-axis?

A: The purpose of reflecting a function over the x-axis is to flip the graph of the function upside down. This can help us understand the behavior of the function and its transformations.

Q: How do I reflect a function over the x-axis?

A: To reflect a function over the x-axis, you need to negate the y-values of the function. This can be done by multiplying the function by -1.

Q: What is the effect of reflecting a function over the x-axis on its range?

A: Reflecting a function over the x-axis will flip the range of the function. If the original function has a range of all real numbers, the reflected function will also have a range of all real numbers.

Q: Can I reflect a function over the x-axis if it is a linear function?

A: Yes, you can reflect a linear function over the x-axis. The reflection of a linear function will also be a linear function.

Q: Can I reflect a function over the x-axis if it is a quadratic function?

A: Yes, you can reflect a quadratic function over the x-axis. The reflection of a quadratic function will also be a quadratic function.

Q: What is the difference between reflecting a function over the x-axis and reflecting a function over the y-axis?

A: Reflecting a function over the x-axis will flip the graph of the function upside down, while reflecting a function over the y-axis will flip the graph of the function left and right.

Q: Can I reflect a function over the x-axis if it is a trigonometric function?

A: Yes, you can reflect a trigonometric function over the x-axis. The reflection of a trigonometric function will also be a trigonometric function.

Q: What is the effect of reflecting a function over the x-axis on its domain?

A: Reflecting a function over the x-axis will not change the domain of the function.

Q: Can I reflect a function over the x-axis if it is a piecewise function?

A: Yes, you can reflect a piecewise function over the x-axis. The reflection of a piecewise function will also be a piecewise function.

Q: What is the difference between reflecting a function over the x-axis and reflecting a function over the y-axis in terms of the graph?

A: Reflecting a function over the x-axis will flip the graph of the function upside down, while reflecting a function over the y-axis will flip the graph of the function left and right.

Conclusion

In this article, we answered some frequently asked questions about reflecting functions over the x-axis. We hope that this article has helped you understand the concept of reflecting functions over the x-axis and its applications.

Final Thoughts

Reflecting functions over the x-axis is an important concept in mathematics that helps us understand the behavior of functions and their transformations. By reflecting a function over the x-axis, we can gain a deeper understanding of the function's behavior and its applications.

Common Mistakes to Avoid

  • Not negating the y-values of the function when reflecting it over the x-axis.
  • Not considering the effect of the reflection on the range of the function.
  • Not understanding the difference between reflecting a function over the x-axis and reflecting a function over the y-axis.

Tips and Tricks

  • Use the concept of reflection to understand the behavior of functions and their transformations.
  • Practice reflecting functions over the x-axis to gain a deeper understanding of the concept.
  • Use the concept of reflection to solve problems and applications in mathematics.