The Function F ( X ) = 3 4 ( 10 ) − X F(x) = \frac{3}{4}(10)^{-x} F ( X ) = 4 3 ​ ( 10 ) − X Is Reflected Across The X X X -axis To Create The Function G ( X G(x G ( X ]. Which Ordered Pair Is On G ( X G(x G ( X ]?A. \left(-3,-\frac{3}{4000}\right ] B. ( − 2 , 75 (-2, 75 ( − 2 , 75 ] C.

Introduction to Reflection Across the $x$-axis

Reflection across the $x$-axis is a fundamental concept in mathematics, particularly in algebra and geometry. When a function is reflected across the $x$-axis, its graph is flipped upside down, resulting in a new function. In this article, we will explore the reflection of the function $f(x) = \frac{3}{4}(10)^{-x}$ across the $x$-axis and determine the ordered pair that lies on the reflected function $g(x)$.

Understanding the Original Function $f(x)$

The original function $f(x) = \frac{3}{4}(10)^{-x}$ is an exponential function with a base of 10 and a coefficient of $\frac{3}{4}$. The function is defined for all real values of $x$ and has a range of positive real numbers. To understand the behavior of the function, let's analyze its components:

• The base of 10 is a positive number, which means that the function will decrease as $x$ increases.
• The coefficient $\frac{3}{4}$ is a positive number, which means that the function will be scaled vertically by a factor of $\frac{3}{4}$.
• The exponent $-x$ indicates that the function will decrease as $x$ increases.

Reflection Across the $x$-axis

When a function is reflected across the $x$-axis, its graph is flipped upside down. This means that the $y$-coordinates of the points on the graph are negated. In other words, if a point $(x, y)$ lies on the original function $f(x)$, then the point $(x, -y)$ lies on the reflected function $g(x)$.

Finding the Ordered Pair on $g(x)$

To find the ordered pair on $g(x)$, we need to find a point $(x, y)$ that lies on the original function $f(x)$ and then negate the $y$-coordinate to get the point on $g(x)$. Let's consider the point $(-2, 75)$, which lies on the original function $f(x)$.

import math
def f(x):
return (3/4) * (10)**(-x)
x = -2
y = f(x)
print(f"Point on f(x): ({x}, {y})")

When we run this code, we get the point $(-2, 75)$, which lies on the original function $f(x)$. To find the point on $g(x)$, we simply negate the $y$-coordinate:

y_g = -y
print(f"Point on g(x): ({x}, {y_g})")

This gives us the point $(-2, -75)$, which lies on the reflected function $g(x)$.

Conclusion

In this article, we explored the reflection of the function $f(x) = \frac{3}{4}(10)^{-x}$ across the $x$-axis and determined the ordered pair that lies on the reflected function $g(x)$. We found that the point $(-2, -75)$ lies on $g(x)$, which is the negation of the point $(-2, 75)$ that lies on the original function $f(x)$. This demonstrates the concept of reflection across the $x$-axis and its application to functions.

Discussion

• What is the significance of reflection across the $x$-axis in mathematics?
• How does the reflection of a function affect its graph?
• Can you think of other examples of functions that can be reflected across the $x$-axis?

Final Answer

The final answer is $\boxed{(-2, -75)}$.

Introduction

In our previous article, we explored the concept of reflection across the $x$-axis and applied it to the function $f(x) = \frac{3}{4}(10)^{-x}$. In this article, we will answer some frequently asked questions about reflection across the $x$-axis and provide additional insights into this important mathematical concept.

Q1: What is reflection across the $x$-axis?

A1: Reflection across the $x$-axis is a transformation that flips a function's graph upside down. When a function is reflected across the $x$-axis, its $y$-coordinates are negated, resulting in a new function.

Q2: How does reflection across the $x$-axis affect the graph of a function?

A2: Reflection across the $x$-axis flips the graph of a function upside down, resulting in a new function with negated $y$-coordinates. This means that if a point $(x, y)$ lies on the original function, the point $(x, -y)$ lies on the reflected function.

Q3: Can you give an example of a function that is reflected across the $x$-axis?

A3: Yes, consider the function $f(x) = 2x^2$. When we reflect this function across the $x$-axis, we get the function $g(x) = -2x^2$. The graph of $g(x)$ is the reflection of the graph of $f(x)$ across the $x$-axis.

Q4: How do you find the reflected function of a given function?

A4: To find the reflected function of a given function, you can simply negate the $y$-coordinates of the points on the original function. For example, if a point $(x, y)$ lies on the original function, the point $(x, -y)$ lies on the reflected function.

Q5: What is the significance of reflection across the $x$-axis in mathematics?

A5: Reflection across the $x$-axis is an important concept in mathematics because it allows us to analyze and understand the behavior of functions in different ways. By reflecting a function across the $x$-axis, we can gain insights into the function's properties and behavior.

Q6: Can you think of any real-world applications of reflection across the $x$-axis?

A6: Yes, reflection across the $x$-axis has many real-world applications. For example, in physics, reflection across the $x$-axis is used to describe the motion of objects. In engineering, reflection across the $x$-axis is used to design and analyze systems.

Q7: How does reflection across the $x$-axis affect the domain and range of a function?

A7: Reflection across the $x$-axis does not affect the domain of a function, but it does affect the range. The range of a function is the set of all possible $y$-values, and when a function is reflected across the $x$-axis, its range is negated.

Q8: Can you give an example of a function that is reflected across the $x$-axis and has a different domain and range?

A8: Yes, consider the function $f(x) = \frac{1}{x}$. When we reflect this function across the $x$-axis, we get the function $g(x) = -\frac{1}{x}$. The domain of $g(x)$ is the same as the domain of $f(x)$, but the range of $g(x)$ is different.

Q9: How does reflection across the $x$-axis affect the graph of a function in terms of its $x$-intercepts?

A9: Reflection across the $x$-axis does not affect the $x$-intercepts of a function. The $x$-intercepts of a function are the points where the function intersects the $x$-axis, and when a function is reflected across the $x$-axis, its $x$-intercepts remain the same.

Q10: Can you give an example of a function that is reflected across the $x$-axis and has the same $x$-intercepts?

A10: Yes, consider the function $f(x) = x^2$. When we reflect this function across the $x$-axis, we get the function $g(x) = -x^2$. The $x$-intercepts of $g(x)$ are the same as the $x$-intercepts of $f(x)$.

Conclusion

In this article, we answered some frequently asked questions about reflection across the $x$-axis and provided additional insights into this important mathematical concept. We hope that this article has been helpful in understanding the concept of reflection across the $x$-axis and its applications in mathematics and real-world scenarios.

Discussion

• What are some other examples of functions that can be reflected across the $x$-axis?
• How does reflection across the $x$-axis affect the graph of a function in terms of its $y$-intercepts?
• Can you think of any other real-world applications of reflection across the $x$-axis?

Final Answer

The final answer is $\boxed{(-2, -75)}$.