Which represents the reflection of [tex]f(x)=\sqrt{x}[/tex] over the [tex]x[/tex]-axis?

The reflection of a function f ( x ) over the x-axis is given by − f ( x ) .
For f ( x ) = x ​ , the reflection is − f ( x ) = − x ​ .
Evaluate − f ( x ) for the given x-values in the tables.
The first table matches the values of − f ( x ) = − x ​ , so the answer is the first table.

Explanation

Understanding the Problem The problem asks us to identify the table that represents the reflection of the function f ( x ) = x ​ over the x-axis.

Finding the Reflected Function When we reflect a function over the x-axis, we are essentially taking the negative of the function's output, so the reflected function will be g ( x ) = − f ( x ) = − x ​ . We need to evaluate this function for the given x-values in the tables and see which table matches the result.

Analyzing the First Table Let's analyze the first table:



When x = − 1 , f ( x ) is undefined because we cannot take the square root of a negative number.
When x = 0 , f ( x ) = − 0 ​ = 0 .
When x = 1 , f ( x ) = − 1 ​ = − 1 .
When x = 4 , f ( x ) = − 4 ​ = − 2 .
This matches the first table.


Analyzing the Second Table Now let's analyze the second table:


When x = − 1 , the table gives f ( x ) = 1 , but f ( x ) should be undefined.
When x = 0 , the table gives f ( x ) = 0 , which is correct.
When x = 1 , the table gives f ( x ) as undefined, but it should be − 1 ​ = − 1 .
When x = 4 , the table gives f ( x ) as undefined, but it should be − 4 ​ = − 2 .
This table does not represent the reflection of f ( x ) over the x-axis.


Conclusion Therefore, the first table represents the reflection of f ( x ) = x ​ over the x-axis.

Examples
Reflections are a fundamental concept in mathematics and physics. For example, when light reflects off a mirror, the angle of incidence equals the angle of reflection. This principle can be modeled mathematically using reflections of functions. Similarly, in computer graphics, reflections are used to create realistic images and animations. Understanding reflections helps in fields ranging from optics to game development.

Answered by GinnyAnswer | 2025-07-06

The reflection of the function f ( x ) = x ​ over the x-axis is g ( x ) = − x ​ . This means that we take the negative output of the original function for each value of x. The resulting function transforms positive outputs into negative ones, representing the reflection.
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Answered by Anonymous | 2025-07-13