To analyze the exponential equation given, \( v(s) = 10 \cdot 2^s \), where \( v \) is the number of views and \( s \) is the number of shares, we can evaluate the statements regarding the domain and range.
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Domain: The variable \( s \) (number of shares) represents the number of times the video has been shared. Since you cannot share a video a negative number of times, the values of \( s \) should be non-negative integers (whole numbers). Therefore, the domain can be represented as \( s \geq 0 \). In terms of real numbers, \( s \) can technically be any non-negative real number.
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Range: The range of the function \( v(s) = 10 \cdot 2^s \):
- When \( s = 0 \), \( v(0) = 10 \cdot 2^0 = 10 \).
- As \( s \) increases, \( v(s) \) increases exponentially.
- Thus, \( v(s) \) approaches infinity as \( s \) increases.
- Therefore, the range of \( v(s) \) includes all values greater than or equal to 10, which means it is \( [10, \infty) \) and does not include 0 or negative numbers.
Now, let's evaluate the provided statements:
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The domain \( s \) values is the real numbers - This statement is false, as the domain should be \( s \geq 0 \) (specifically, non-negative real numbers).
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The range values includes the values greater than 0 - This statement is true, as the range indeed includes all values starting from 10 and going to infinity.
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The range values is all real numbers - This statement is false, since the range does not include any values less than 10.
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The domain \( s \) values include the values \( s \) is less than or equal to - This statement is incomplete and ambiguous. If it implies that \( s \) can be any negative number or specifically just "less than or equal to" without specifying a number, it is false, since valid values for \( s \) start from 0.
The best answer that fits among the statements is: The range values includes the values greater than 0.