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Sagot :
Let's analyze the function [tex]\( f(x) = 49 \left(\frac{1}{7}\right)^x \)[/tex] to determine the correct options.
### Domain Analysis
1. The domain is the set of all real numbers.
The expression [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions on [tex]\( x \)[/tex], so the domain of the function is indeed the set of all real numbers.
### Range Analysis
2. The range is the set of all real numbers.
The base [tex]\( \frac{1}{7} \)[/tex] is a positive fraction less than 1. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] becomes very large, and as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] approaches zero. Because the function multiplies [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] by 49, [tex]\( f(x) \)[/tex] will never be zero or negative but will cover all positive values.
3. The range is [tex]\( y > 0 \)[/tex].
Since [tex]\( 49 \left( \frac{1}{7} \right)^x \)[/tex] yields positive values for all real [tex]\( x \)[/tex], the range is indeed [tex]\( y > 0 \)[/tex].
### Domain Constraint
4. The domain is [tex]\( x > 0 \)[/tex].
As previously analyzed, there are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex]. Therefore, stating that the domain is [tex]\( x > 0 \)[/tex] is incorrect.
### Function Behavior
5. As [tex]\( x \)[/tex] increases by 1, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value.
Notice that [tex]\( f(x+1) = 49 \left( \frac{1}{7} \right)^{x+1} = 49 \left( \frac{1}{7} \right)^x \cdot \frac{1}{7} = \frac{1}{7} \cdot 49 \left( \frac{1}{7} \right)^x = \frac{1}{7} f(x) \)[/tex]. Thus, as [tex]\( x \)[/tex] increases by 1, the new [tex]\( y \)[/tex]-value is indeed one-seventh of the previous [tex]\( y \)[/tex]-value.
### Conclusion
From the analysis, the following three statements are true:
1. The domain is the set of all real numbers.
2. The range is [tex]\( y > 0 \)[/tex].
3. As [tex]\( x \)[/tex] increases by 1, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value.
### Domain Analysis
1. The domain is the set of all real numbers.
The expression [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions on [tex]\( x \)[/tex], so the domain of the function is indeed the set of all real numbers.
### Range Analysis
2. The range is the set of all real numbers.
The base [tex]\( \frac{1}{7} \)[/tex] is a positive fraction less than 1. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] becomes very large, and as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] approaches zero. Because the function multiplies [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] by 49, [tex]\( f(x) \)[/tex] will never be zero or negative but will cover all positive values.
3. The range is [tex]\( y > 0 \)[/tex].
Since [tex]\( 49 \left( \frac{1}{7} \right)^x \)[/tex] yields positive values for all real [tex]\( x \)[/tex], the range is indeed [tex]\( y > 0 \)[/tex].
### Domain Constraint
4. The domain is [tex]\( x > 0 \)[/tex].
As previously analyzed, there are no restrictions on [tex]\( x \)[/tex] for the function [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex]. Therefore, stating that the domain is [tex]\( x > 0 \)[/tex] is incorrect.
### Function Behavior
5. As [tex]\( x \)[/tex] increases by 1, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value.
Notice that [tex]\( f(x+1) = 49 \left( \frac{1}{7} \right)^{x+1} = 49 \left( \frac{1}{7} \right)^x \cdot \frac{1}{7} = \frac{1}{7} \cdot 49 \left( \frac{1}{7} \right)^x = \frac{1}{7} f(x) \)[/tex]. Thus, as [tex]\( x \)[/tex] increases by 1, the new [tex]\( y \)[/tex]-value is indeed one-seventh of the previous [tex]\( y \)[/tex]-value.
### Conclusion
From the analysis, the following three statements are true:
1. The domain is the set of all real numbers.
2. The range is [tex]\( y > 0 \)[/tex].
3. As [tex]\( x \)[/tex] increases by 1, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value.
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