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A function of the form [tex]f(x) = a b^x[/tex] is modified so that the [tex]b[/tex] value remains the same but the [tex]a[/tex] value is increased by 2. How do the domain and range of the new function compare to the domain and range of the original function? Check all that apply.

- The range stays the same.
- The range becomes [tex]y \ \textgreater \ 2[/tex].
- The domain stays the same.
- The domain becomes [tex]x \ \textgreater \ 2[/tex].
- The range becomes [tex]y \geq 2[/tex].
- The domain becomes [tex]x \geq 2[/tex].

Sagot :

GinnyAnsweridn
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]
```
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