Question
The view of a particular video are increasing at a rate of 100% for each share the video receives. If the video originated with 10 views the situation can be modeled by the exponential equation v(s) = 19(2)^s where v is the number of views for every s shares. Which of the following statements is true about the equation
The range (v-values) is all real but
The domain (s-values) is all real numbers
The range (v-values) includes the values greater than (0< v <=10)
The domain (s-values) includes the values (s>=o)
The range (v-values) is all real but
The domain (s-values) is all real numbers
The range (v-values) includes the values greater than (0< v <=10)
The domain (s-values) includes the values (s>=o)
Answers
Answered by
GPT-4o mini
Let's analyze the given information about the exponential equation \( v(s) = 19(2)^s \).
1. **Domain (s-values)**:
- The variable \( s \) represents the number of shares. In real-world scenarios, you cannot have a negative number of shares, so \( s \) must be greater than or equal to 0. Therefore, the domain is:
\[
\text{Domain: } s \geq 0
\]
2. **Range (v-values)**:
- The function \( v(s) = 19(2)^s \) describes an exponential growth model. As \( s \) increases, \( v(s) \) also increases without bound. Even when \( s = 0 \), \( v(0) = 19(2)^0 = 19 \).
- Thus, the minimum value of \( v \) is 19 when \( s = 0 \), and as \( s \) approaches infinity, \( v \) approaches infinity as well. Therefore, the range of \( v(s) \) is:
\[
\text{Range: } v \geq 19
\]
3. **Evaluating the statements**:
- "The range (v-values) is all real but the domain (s-values) is all real numbers." - **False** (It's not all real for the range, as values are \( v \geq 19 \)).
- "The range (v-values) includes the values greater than (0 < v <= 10)." - **False** (The range starts at 19, so it doesn't include any values less than 19).
- "The domain (s-values) includes the values (s >= 0)." - **True** (This is correct since \( s \) cannot be negative).
- "The range (v-values) is all real numbers." - **False** (Again, because the function only takes on values greater than or equal to 19).
Based on this analysis, the only true statement is:
**The domain (s-values) includes the values (s >= 0).**
1. **Domain (s-values)**:
- The variable \( s \) represents the number of shares. In real-world scenarios, you cannot have a negative number of shares, so \( s \) must be greater than or equal to 0. Therefore, the domain is:
\[
\text{Domain: } s \geq 0
\]
2. **Range (v-values)**:
- The function \( v(s) = 19(2)^s \) describes an exponential growth model. As \( s \) increases, \( v(s) \) also increases without bound. Even when \( s = 0 \), \( v(0) = 19(2)^0 = 19 \).
- Thus, the minimum value of \( v \) is 19 when \( s = 0 \), and as \( s \) approaches infinity, \( v \) approaches infinity as well. Therefore, the range of \( v(s) \) is:
\[
\text{Range: } v \geq 19
\]
3. **Evaluating the statements**:
- "The range (v-values) is all real but the domain (s-values) is all real numbers." - **False** (It's not all real for the range, as values are \( v \geq 19 \)).
- "The range (v-values) includes the values greater than (0 < v <= 10)." - **False** (The range starts at 19, so it doesn't include any values less than 19).
- "The domain (s-values) includes the values (s >= 0)." - **True** (This is correct since \( s \) cannot be negative).
- "The range (v-values) is all real numbers." - **False** (Again, because the function only takes on values greater than or equal to 19).
Based on this analysis, the only true statement is:
**The domain (s-values) includes the values (s >= 0).**
Answer
The equation is v(s) = 10(2)^s
Answered by
GPT-4o mini
Let's analyze the updated exponential equation \( v(s) = 10(2)^s \).
1. **Domain (s-values)**:
- The variable \( s \) represents the number of shares. In a real-world context, the number of shares cannot be negative, so \( s \) must be greater than or equal to 0. Therefore, the domain is:
\[
\text{Domain: } s \geq 0
\]
2. **Range (v-values)**:
- The function \( v(s) = 10(2)^s \) describes an exponential growth. At \( s = 0 \):
\[
v(0) = 10(2)^0 = 10
\]
- As \( s \) increases, \( v(s) \) increases without bound as \( (2)^s \) grows. Thus, the minimum value of \( v(s) \) is 10 when \( s = 0 \), and it approaches infinity as \( s \) increases. Therefore, the range of \( v(s) \) is:
\[
\text{Range: } v \geq 10
\]
3. **Evaluating the statements**:
- "The range (v-values) is all real but the domain (s-values) is all real numbers." - **False** – The range is not all real (it starts at 10 and goes to infinity).
- "The range (v-values) includes the values greater than (0 < v <= 10)." - **False** – The range starts at 10, so it does not include values less than or equal to 10.
- "The domain (s-values) includes the values (s >= 0)." - **True** – This statement is correct since \( s \) cannot be negative.
- "The range (v-values) is all real numbers." - **False** – The range is limited to values \( v \geq 10\).
Based on this analysis, the only true statement is:
**The domain (s-values) includes the values (s >= 0).**
1. **Domain (s-values)**:
- The variable \( s \) represents the number of shares. In a real-world context, the number of shares cannot be negative, so \( s \) must be greater than or equal to 0. Therefore, the domain is:
\[
\text{Domain: } s \geq 0
\]
2. **Range (v-values)**:
- The function \( v(s) = 10(2)^s \) describes an exponential growth. At \( s = 0 \):
\[
v(0) = 10(2)^0 = 10
\]
- As \( s \) increases, \( v(s) \) increases without bound as \( (2)^s \) grows. Thus, the minimum value of \( v(s) \) is 10 when \( s = 0 \), and it approaches infinity as \( s \) increases. Therefore, the range of \( v(s) \) is:
\[
\text{Range: } v \geq 10
\]
3. **Evaluating the statements**:
- "The range (v-values) is all real but the domain (s-values) is all real numbers." - **False** – The range is not all real (it starts at 10 and goes to infinity).
- "The range (v-values) includes the values greater than (0 < v <= 10)." - **False** – The range starts at 10, so it does not include values less than or equal to 10.
- "The domain (s-values) includes the values (s >= 0)." - **True** – This statement is correct since \( s \) cannot be negative.
- "The range (v-values) is all real numbers." - **False** – The range is limited to values \( v \geq 10\).
Based on this analysis, the only true statement is:
**The domain (s-values) includes the values (s >= 0).**