Let's analyze the function [tex]\( f(x) = ab^x \)[/tex] and see how it changes when we increase the value of [tex]\( a \)[/tex] by 2.
### Step-by-Step Analysis:
1. Original Function [tex]\( f(x) = ab^x \)[/tex]:
- Domain: The domain of the function [tex]\( f(x) = ab^x \)[/tex] is all real numbers, [tex]\( x \in \mathbb{R} \)[/tex]. For any value of [tex]\( x \)[/tex], [tex]\( b^x \)[/tex] is defined and positive if [tex]\( b \)[/tex] is positive.
- Range: Since [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are positive constants and [tex]\( b^x \)[/tex] is always positive, the function [tex]\( ab^x \)[/tex] will also always be positive. Therefore, the range of the function is [tex]\( y > 0 \)[/tex].
2. Modified Function [tex]\( g(x) = (a+2)b^x \)[/tex]:
- We increase the value of [tex]\( a \)[/tex] by 2, resulting in the new function [tex]\( g(x) = (a+2)b^x \)[/tex].
- Domain: The domain remains the same because the modifications to the coefficient [tex]\( a \)[/tex] do not affect the values [tex]\( x \)[/tex] can take. The domain of [tex]\( g(x) \)[/tex], like [tex]\( f(x) \)[/tex], is all real numbers, [tex]\( x \in \mathbb{R} \)[/tex].
- Range: The range will shift due to the increase in [tex]\( a \)[/tex]. Adding 2 to [tex]\( a \)[/tex] shifts the range upwards. Originally, the range of [tex]\( f(x) = ab^x \)[/tex] was [tex]\( y > 0 \)[/tex]. Adding 2 to the coefficient [tex]\( a \)[/tex] means the function now starts from a higher minimum value. Specifically, the range of [tex]\( g(x) = (a+2)b^x \)[/tex] becomes [tex]\( y > 2 \)[/tex].
### Conclusion:
From our analysis, we can conclude the following changes:
- The domain remains the same. Thus, "The domain stays the same" is a correct statement.
- The range shifts such that it starts from [tex]\( y > 2 \)[/tex]. Hence, "The range becomes [tex]\( y > 2 \)[/tex]" is a correct statement.
Therefore, the valid statements are:
- The range becomes [tex]\( y > 2 \)[/tex].
- The domain stays the same.
### Final Answer:
The valid statements in this case are:
- The range becomes [tex]\( y > 2 \)[/tex].
- The domain stays the same.
So, the correct items to check are:
1. The range becomes [tex]\( y > 2 \)[/tex].
2. The domain stays the same.
Thus, the correct selections are:
```
[2, 3]
```
