Which represents the reflection of [tex]f(x)=\sqrt{x}[/tex] over the [tex]y[/tex]-axis?
| x | f(x) |
| --- | --------- |
| -1 | undefined |
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |
| x | f(x) |
| --- | ------ |
| -1 | 1 |
| 0 | 0 |
| 1 | -1 |
| 4 | -2 |
Answer (2)
To reflect f ( x ) = x over the y-axis, replace x with − x , resulting in f ( − x ) = − x .
Evaluate − x for x = − 1 , 0 , 1 , 4 .
Compare the calculated values with the values in the second table.
The second table does not correctly represent the reflection because it contains incorrect values; the square root of a number cannot be negative, and the square root of a negative number is undefined in the real number system. Therefore, the second table is not the reflection of the given function. The correct reflection would have f ( 1 ) and f ( 4 ) undefined.
Explanation
Finding the Reflection To find the reflection of a function f ( x ) over the y -axis, we replace x with − x . So, if f ( x ) = x , the reflection over the y -axis is f ( − x ) = − x . This means we need to evaluate − x for different values of x and compare the results with the given table.
Analyzing the Original Function Let's analyze the given table for the original function f ( x ) = x :
When x = − 1 , f ( x ) is undefined because we cannot take the square root of a negative number in the real number system.
When x = 0 , f ( 0 ) = 0 = 0 .
When x = 1 , f ( 1 ) = 1 = 1 .
When x = 4 , f ( 4 ) = 4 = 2 .
Analyzing the Reflection Now, let's analyze the second table to see if it represents the reflection f ( − x ) = − x :
When x = − 1 , f ( − ( − 1 )) = f ( 1 ) = − ( − 1 ) = 1 = 1 . This matches the second table.
When x = 0 , f ( − 0 ) = f ( 0 ) = − 0 = 0 = 0 . This matches the second table.
When x = 1 , f ( − 1 ) = − 1 . This is undefined in the real number system. However, the second table gives a value of − 1 , which is incorrect. Note that the square root function always returns non-negative values.
When x = 4 , f ( − 4 ) = − 4 . This is also undefined in the real number system. However, the second table gives a value of − 2 , which is incorrect.
Conclusion Since the second table does not correctly represent the reflection of f ( x ) = x over the y -axis, as it contains incorrect values for x = 1 and x = 4 , the second table is not the correct representation of the reflection.
Examples
Reflections are used in physics to study the behavior of light and other waves when they encounter a reflective surface. For example, the path of a light ray reflecting off a mirror can be modeled using reflections. In computer graphics, reflections are used to create realistic images of objects in a scene. Understanding reflections helps in designing optical instruments and creating visual effects in movies and video games.
The reflection of f ( x ) = x over the y -axis results in f ( − x ) = − x . The second table fails to represent this reflection accurately as it lists undefined values incorrectly. Thus, it does not correctly show the reflection of the function.
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