A function and its reflection can be identified by the following equations: f(x)=x−2
and f′(x)=−2−x
. Determine the line of reflection. Enter the option number of your answer.
Option #1: the x-axis
Option #2: the y-axis
Option #3: y=x
Option #4: y=−x
(1 point)
The line of reflection is Option #
Option #2: the y-axis
how
To determine the line of reflection, we need to identify when the function and its reflection are the same.
Let's compare the given equation of the function, f(x)=x−2, to the equation of its reflection, f'(x)=−2−x.
We can see that the reflection is obtained by changing the sign of the x-coefficient. In this case, the original x-coefficient is 1, and its reflection has a coefficient of -1.
Since the x-axis is the line that separates positive and negative values of y, and changing the sign of the x-coefficient reflects the function across the x-axis, the line of reflection is Option #1: the x-axis.
To determine the line of reflection, we need to examine the relationship between the original function f(x) = x - 2 and its reflection f'(x) = -2 - x.
We know that the reflection of a function occurs when we replace x with -x. So, let's replace x with -x in the original function f(x) = x - 2:
f(-x) = -x - 2
Comparing this reflection f'(-x) = -2 - x with f(x), we can see that they are equivalent. This implies that the line of reflection is the y-axis.
Therefore, the correct option is Option #2: the y-axis.