The domain of the transformed function (reflected across the y-axis and translated left 5 units) is the same as the parent function, but their ranges differ due to the reflection.
When the graph of
is reflected across the y-axis and translated to the left 5 units, it undergoes two transformations. Let's analyze the effects of each transformation on the domain and range:
1. **Reflection across the y-axis:** This transformation negates the x-values. For any value of \(x\) in the original function, the reflected function will have \(-x\). This means the domain is affected.
2. **Translation to the left 5 units:** This transformation shifts the graph horizontally to the left. If
represents the original function, the transformed function is
. This means the domain is affected, but the range remains the same.
Now, let's consider the options:
- Both the domain and range of the transformed function are the same as those of the parent function: This is not true because the reflection and translation affect the domain.
- Neither the domain nor the range of the transformed function are the same as those of the parent function: This is not entirely true because the range remains the same.
- The range of the transformed function is the same as the parent function, but the domains of the functions are different: This is not true because both the domain and range are affected.
- The domain of the transformed function is the same as the parent function, but the ranges of the functions are different: This is the correct statement. The domain is the same because the translation does not affect it, but the range is different due to the reflection across the y-axis.
Therefore, the correct statement is: **The domain of the transformed function is the same as the parent function, but the ranges of the functions are different.**