The Parent Function Is $f(x)=\sqrt{x}$, And A Transformation Of The Parent Function, $g(x$\], Is A Reflection Over The $x$-axis. Write The Function $g(x$\].$g(x) = $

Introduction

In mathematics, a parent function is a basic function that has been transformed in various ways to create new functions. The parent function $f(x)=\sqrt{x}$ is a fundamental function in mathematics, and its transformations are essential in understanding various mathematical concepts. In this article, we will discuss the transformation of the parent function $f(x)=\sqrt{x}$ to create a new function $g(x)$, which is a reflection over the $x$-axis.

Understanding the Parent Function

The parent function $f(x)=\sqrt{x}$ is a square root function that takes a non-negative value as input and returns the square root of that value. This function is defined for all non-negative real numbers, and its graph is a curve that opens upwards. The parent function $f(x)=\sqrt{x}$ has several important properties, including:

• Domain: The domain of the parent function $f(x)=\sqrt{x}$ is all non-negative real numbers, i.e., $x \geq 0$.
• Range: The range of the parent function $f(x)=\sqrt{x}$ is all non-negative real numbers, i.e., $y \geq 0$.
• Graph: The graph of the parent function $f(x)=\sqrt{x}$ is a curve that opens upwards.

Reflection Over the X-Axis

A reflection over the $x$-axis is a transformation that flips a function over the $x$-axis. This transformation is denoted by a negative sign in front of the function. In other words, if we have a function $f(x)$, then its reflection over the $x$-axis is given by $-f(x)$.

Writing the Function g(x)

To write the function $g(x)$, which is a reflection over the $x$-axis, we need to multiply the parent function $f(x)=\sqrt{x}$ by $-1$. This is because the reflection over the $x$-axis is denoted by a negative sign in front of the function.

Therefore, the function $g(x)$ is given by:

$$g(x) = -f(x) = -\sqrt{x}$$

Properties of the Function g(x)

The function $g(x) = -\sqrt{x}$ has several important properties, including:

• Domain: The domain of the function $g(x) = -\sqrt{x}$ is all non-negative real numbers, i.e., $x \geq 0$.
• Range: The range of the function $g(x) = -\sqrt{x}$ is all non-positive real numbers, i.e., $y \leq 0$.
• Graph: The graph of the function $g(x) = -\sqrt{x}$ is a curve that opens downwards.

Conclusion

In conclusion, the function $g(x) = -\sqrt{x}$ is a reflection over the $x$-axis of the parent function $f(x) = \sqrt{x}$. This function has several important properties, including its domain, range, and graph. Understanding the transformations of the parent function is essential in mathematics, and this article has provided a detailed explanation of the reflection over the $x$-axis.

Example Problems

Here are some example problems that illustrate the concept of reflection over the $x$-axis:

• Problem 1: Find the reflection over the $x$-axis of the function $f(x) = 2x^2$.
• Problem 2: Find the reflection over the $x$-axis of the function $f(x) = \sin x$.
• Problem 3: Find the reflection over the $x$-axis of the function $f(x) = \cos x$.

Solutions

Here are the solutions to the example problems:

• Problem 1: The reflection over the $x$-axis of the function $f(x) = 2x^2$ is given by $-f(x) = -2x^2$.
• Problem 2: The reflection over the $x$-axis of the function $f(x) = \sin x$ is given by $-f(x) = -\sin x$.
• Problem 3: The reflection over the $x$-axis of the function $f(x) = \cos x$ is given by $-f(x) = -\cos x$.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the reflection over the $x$-axis.

Q: What is a reflection over the x-axis?

A: A reflection over the $x$-axis is a transformation that flips a function over the $x$-axis. This transformation is denoted by a negative sign in front of the function.

Q: How do I write the function g(x) after a reflection over the x-axis?

A: To write the function $g(x)$ after a reflection over the $x$-axis, you need to multiply the parent function $f(x)$ by $-1$. This is because the reflection over the $x$-axis is denoted by a negative sign in front of the function.

Q: What are the properties of the function g(x) after a reflection over the x-axis?

A: The function $g(x)$ after a reflection over the $x$-axis has several important properties, including:

• Domain: The domain of the function $g(x)$ is the same as the domain of the parent function $f(x)$.
• Range: The range of the function $g(x)$ is the negative of the range of the parent function $f(x)$.
• Graph: The graph of the function $g(x)$ is the negative of the graph of the parent function $f(x)$.

Q: How do I find the reflection over the x-axis of a function?

A: To find the reflection over the $x$-axis of a function, you need to multiply the function by $-1$. This is because the reflection over the $x$-axis is denoted by a negative sign in front of the function.

Q: What are some examples of functions that have been reflected over the x-axis?

A: Some examples of functions that have been reflected over the $x$-axis include:

• Reflection of f(x) = x^2: The reflection of $f(x) = x^2$ over the $x$-axis is $-f(x) = -x^2$.
• Reflection of f(x) = sin(x): The reflection of $f(x) = \sin(x)$ over the $x$-axis is $-f(x) = -\sin(x)$.
• Reflection of f(x) = cos(x): The reflection of $f(x) = \cos(x)$ over the $x$-axis is $-f(x) = -\cos(x)$.

Q: What are some real-world applications of reflections over the x-axis?

A: Some real-world applications of reflections over the $x$-axis include:

• Physics: Reflections over the $x$-axis are used to describe the motion of objects in physics.
• Engineering: Reflections over the $x$-axis are used to design and analyze systems in engineering.
• Computer Science: Reflections over the $x$-axis are used in computer graphics and game development.

Q: How do I graph a function after a reflection over the x-axis?

A: To graph a function after a reflection over the $x$-axis, you need to reflect the graph of the parent function over the $x$-axis. This can be done by multiplying the function by $-1$ and then graphing the resulting function.

Conclusion

In conclusion, reflections over the $x$-axis are an essential concept in mathematics that helps us understand various mathematical concepts. The function $g(x) = -\sqrt{x}$ is a reflection over the $x$-axis of the parent function $f(x) = \sqrt{x}$, and it has several important properties, including its domain, range, and graph. Understanding the transformations of the parent function is essential in mathematics, and this article has provided a detailed explanation of the reflection over the $x$-axis.