Question
Henry uses the equation m(x) = 12,450 (0.83)^x, where x represents the number of years after purchase, to model the value of some farm equipment for the first 6 years.
Which TWO statements must be TRUE?
The value of m (1) is 12,450.
m (6) can have multiple values Y = M(x)
The range for the given domain is (-infinity,12,450)
m represents a function.
Which TWO statements must be TRUE?
The value of m (1) is 12,450.
m (6) can have multiple values Y = M(x)
The range for the given domain is (-infinity,12,450)
m represents a function.
Answers
Answered by
GPT-4o mini
To analyze the given equation \( m(x) = 12,450 (0.83)^x \) and determine which statements must be true, let's go through them one by one.
1. **The value of m(1) is 12,450.**
- If we calculate \( m(1) \):
\[
m(1) = 12,450 (0.83)^1 = 12,450 \times 0.83 = 10,351.50
\]
- Therefore, this statement is **false**.
2. **m(6) can have multiple values Y = M(x).**
- The function \( m(x) = 12,450 (0.83)^x \) is a single-valued function for each \( x \). This means that for any specific \( x \), there is only one value of \( m(x) \). Therefore, \( m(6) \) will have only one value, and this statement is **false**.
3. **The range for the given domain is (-∞, 12,450).**
- The function \( m(x) \) is an exponential decay function since \( 0.83 < 1 \). As \( x \) increases, \( m(x) \) will approach zero but never actually reach it. The maximum value of the function occurs at \( x = 0 \), where \( m(0) = 12,450 \). The range of the function is therefore \( (0, 12,450] \), not \( (-\infty, 12,450) \). So, this statement is **false**.
4. **m represents a function.**
- The definition of a function states that for each input \( x \), there is exactly one output \( m(x) \). Since \( m(x) = 12,450 (0.83)^x \) gives one and only one output for every value of \( x \), it meets the criteria for a function. Therefore, this statement is **true**.
Based on the analysis:
- The only true statement from the options provided is:
- **m represents a function.**
Thus, the two statements that must be true are:
1. **m represents a function.**
2. (There is no second true statement; hence only one statement is definitively true according to the options you've given.)
1. **The value of m(1) is 12,450.**
- If we calculate \( m(1) \):
\[
m(1) = 12,450 (0.83)^1 = 12,450 \times 0.83 = 10,351.50
\]
- Therefore, this statement is **false**.
2. **m(6) can have multiple values Y = M(x).**
- The function \( m(x) = 12,450 (0.83)^x \) is a single-valued function for each \( x \). This means that for any specific \( x \), there is only one value of \( m(x) \). Therefore, \( m(6) \) will have only one value, and this statement is **false**.
3. **The range for the given domain is (-∞, 12,450).**
- The function \( m(x) \) is an exponential decay function since \( 0.83 < 1 \). As \( x \) increases, \( m(x) \) will approach zero but never actually reach it. The maximum value of the function occurs at \( x = 0 \), where \( m(0) = 12,450 \). The range of the function is therefore \( (0, 12,450] \), not \( (-\infty, 12,450) \). So, this statement is **false**.
4. **m represents a function.**
- The definition of a function states that for each input \( x \), there is exactly one output \( m(x) \). Since \( m(x) = 12,450 (0.83)^x \) gives one and only one output for every value of \( x \), it meets the criteria for a function. Therefore, this statement is **true**.
Based on the analysis:
- The only true statement from the options provided is:
- **m represents a function.**
Thus, the two statements that must be true are:
1. **m represents a function.**
2. (There is no second true statement; hence only one statement is definitively true according to the options you've given.)