Let's analyze the transformations involved and how they affect the domain and range of the given function [tex]\( f(x) = |x| \)[/tex].
1. Reflection Across the X-Axis:
- The original function [tex]\( f(x) = |x| \)[/tex] outputs absolute values of [tex]\( x \)[/tex]. By reflecting this graph across the x-axis, each positive output is turned into its negative counterpart.
- The new function after reflection becomes [tex]\( f(x) = -|x| \)[/tex].
2. Translation to the Right by 6 Units:
- Translating the function [tex]\( f(x) = -|x| \)[/tex] to the right by 6 units means shifting the graph horizontally to the right.
- The new function after this transformation is [tex]\( f(x) = -|x - 6| \)[/tex].
Now let's consider the domain and range before and after these transformations:
- Domain (Original Function [tex]\( f(x) = |x| \)[/tex]):
- The domain of [tex]\( f(x) = |x| \)[/tex] is all real numbers because you can input any real number into the absolute value function.
- Thus, the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range (Original Function [tex]\( f(x) = |x| \)[/tex]):
- The range of [tex]\( f(x) = |x| \)[/tex] is all non-negative real numbers because the absolute value of any real number is always non-negative.
- Thus, the range is [tex]\( [0, \infty) \)[/tex].
- Domain (Transformed Function [tex]\( f(x) = -|x - 6| \)[/tex]):
- The domain of the transformed function remains all real numbers since neither reflecting about the x-axis nor translating right affects the set of possible [tex]\( x \)[/tex]-values we can input into the function.
- Thus, the domain is still [tex]\( (-\infty, \infty) \)[/tex].
- Range (Transformed Function [tex]\( f(x) = -|x - 6| \)[/tex]):
- Reflecting the graph across the x-axis changes all positive outputs to negative outputs. Initially, [tex]\( |x| \)[/tex] gives outputs in [tex]\( [0, \infty) \)[/tex], but now [tex]\( -|x| \)[/tex] will give outputs in [tex]\( (-\infty, 0] \)[/tex].
- Translating the graph horizontally does not affect the range as it does not alter the [tex]\( y \)[/tex]-values themselves, just their associated [tex]\( x \)[/tex]-coordinates.
- Thus, the range of the transformed function is [tex]\( (-\infty, 0] \)[/tex].
Therefore, the correct statement about the domain and range of the transformed function compared to the parent function is:
- The domain of the transformed function is the same as the parent function, but the ranges of the functions are different.
