Answer :
Let's analyze the function [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex] in detail:
1. Domain Analysis:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- In this case, the function [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- Therefore, the domain is the set of all real numbers.
2. Range Analysis:
- The range of a function is the set of all possible output values (y-values).
- Here, [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex]. The base [tex]\( \left( \frac{1}{7} \right) \)[/tex] is a positive fraction less than 1.
- As [tex]\( x \)[/tex] takes any real number, [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] will always be positive because any positive fraction raised to any power remains positive.
- Multiplying this positive value by 49, which is also positive, results in output values [tex]\( y \)[/tex] that are always positive.
- Thus, the range of the function is [tex]\( y > 0 \)[/tex].
3. Specific Domain Assertion:
- It is stated that the domain is [tex]\( x > 0 \)[/tex]. However, there are no restrictions on [tex]\( x \)[/tex], and [tex]\( x \)[/tex] can be any real number.
- Therefore, the domain being [tex]\( x > 0 \)[/tex] is not true.
4. Specific Range Assertion:
- It is stated that the range is the set of all real numbers.
- However, we have established that the range of the function is [tex]\( y > 0 \)[/tex] because the outputs are always positive and never zero or negative.
- So, the range being all real numbers is not true.
5. Behavior of [tex]\( y \)[/tex] as [tex]\( x \)[/tex] Increases:
- As [tex]\( x \)[/tex] increases by 1, consider [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex].
- If [tex]\( x \)[/tex] increases to [tex]\( x+1 \)[/tex], we get [tex]\( f(x+1) = 49 \left( \frac{1}{7} \right)^{x+1} = 49 \left( \frac{1}{7} \right)^x \cdot \left( \frac{1}{7} \right) = \left( 49 \left( \frac{1}{7} \right)^x \right) \cdot \frac{1}{7} = f(x) \cdot \frac{1}{7} \)[/tex].
- Thus, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value as [tex]\( x \)[/tex] increases by 1.
Based on this detailed analysis:
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.
Therefore, the three true statements are:
- The domain is the set of all real numbers.
- The range is [tex]\( y > 0 \)[/tex].
- As [tex]\( x \)[/tex] increases by 1, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value.
1. Domain Analysis:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- In this case, the function [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- Therefore, the domain is the set of all real numbers.
2. Range Analysis:
- The range of a function is the set of all possible output values (y-values).
- Here, [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex]. The base [tex]\( \left( \frac{1}{7} \right) \)[/tex] is a positive fraction less than 1.
- As [tex]\( x \)[/tex] takes any real number, [tex]\( \left( \frac{1}{7} \right)^x \)[/tex] will always be positive because any positive fraction raised to any power remains positive.
- Multiplying this positive value by 49, which is also positive, results in output values [tex]\( y \)[/tex] that are always positive.
- Thus, the range of the function is [tex]\( y > 0 \)[/tex].
3. Specific Domain Assertion:
- It is stated that the domain is [tex]\( x > 0 \)[/tex]. However, there are no restrictions on [tex]\( x \)[/tex], and [tex]\( x \)[/tex] can be any real number.
- Therefore, the domain being [tex]\( x > 0 \)[/tex] is not true.
4. Specific Range Assertion:
- It is stated that the range is the set of all real numbers.
- However, we have established that the range of the function is [tex]\( y > 0 \)[/tex] because the outputs are always positive and never zero or negative.
- So, the range being all real numbers is not true.
5. Behavior of [tex]\( y \)[/tex] as [tex]\( x \)[/tex] Increases:
- As [tex]\( x \)[/tex] increases by 1, consider [tex]\( f(x) = 49 \left( \frac{1}{7} \right)^x \)[/tex].
- If [tex]\( x \)[/tex] increases to [tex]\( x+1 \)[/tex], we get [tex]\( f(x+1) = 49 \left( \frac{1}{7} \right)^{x+1} = 49 \left( \frac{1}{7} \right)^x \cdot \left( \frac{1}{7} \right) = \left( 49 \left( \frac{1}{7} \right)^x \right) \cdot \frac{1}{7} = f(x) \cdot \frac{1}{7} \)[/tex].
- Thus, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value as [tex]\( x \)[/tex] increases by 1.
Based on this detailed analysis:
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.
Therefore, the three true statements are:
- The domain is the set of all real numbers.
- The range is [tex]\( y > 0 \)[/tex].
- As [tex]\( x \)[/tex] increases by 1, each [tex]\( y \)[/tex]-value is one-seventh of the previous [tex]\( y \)[/tex]-value.
