Answer :
To understand how the function [tex]\( f(x) = 603(1.3)^x \)[/tex] behaves, we need to analyze its structure and what each component represents.
1. Function Analysis:
- The function [tex]\( f(x) \)[/tex] describes the number of students enrolled at a university [tex]\( x \)[/tex] years after it was founded.
- The term [tex]\( 603 \)[/tex] is a constant multiplier, representing the initial number of students when [tex]\( x = 0 \)[/tex].
- The term [tex]\( (1.3)^x \)[/tex] indicates how the number of students changes each year. Specifically, [tex]\( 1.3 \)[/tex] is used as the base of the exponent.
2. Understanding the Growth Factor:
- In exponential functions of the form [tex]\( f(x) = a \cdot b^x \)[/tex], [tex]\( a \)[/tex] is the initial value, and [tex]\( b \)[/tex] is the base of the exponent, known as the growth factor.
- The base [tex]\( b = 1.3 \)[/tex] in the function [tex]\( f(x) = 603(1.3)^x \)[/tex] indicates that each year, the number of students is multiplied by 1.3 compared to the previous year.
3. Interpreting the Growth Factor:
- A growth factor of [tex]\( 1.3 \)[/tex] means that each year, the new number of students is 1.3 times the number of students from the previous year.
- This means there is a 30% increase annually, but the best way to describe this increase in terms of a specific quantity is that each year the number of students is multiplied by 1.3.
Therefore, the statement "Each year, the number of students is [tex]\( \_\_\_\_\_\_\_\_ \)[/tex] the number the year before" can best be completed by:
A. 1.3 times
Thus, the correct answer is A. 1.3 times.
1. Function Analysis:
- The function [tex]\( f(x) \)[/tex] describes the number of students enrolled at a university [tex]\( x \)[/tex] years after it was founded.
- The term [tex]\( 603 \)[/tex] is a constant multiplier, representing the initial number of students when [tex]\( x = 0 \)[/tex].
- The term [tex]\( (1.3)^x \)[/tex] indicates how the number of students changes each year. Specifically, [tex]\( 1.3 \)[/tex] is used as the base of the exponent.
2. Understanding the Growth Factor:
- In exponential functions of the form [tex]\( f(x) = a \cdot b^x \)[/tex], [tex]\( a \)[/tex] is the initial value, and [tex]\( b \)[/tex] is the base of the exponent, known as the growth factor.
- The base [tex]\( b = 1.3 \)[/tex] in the function [tex]\( f(x) = 603(1.3)^x \)[/tex] indicates that each year, the number of students is multiplied by 1.3 compared to the previous year.
3. Interpreting the Growth Factor:
- A growth factor of [tex]\( 1.3 \)[/tex] means that each year, the new number of students is 1.3 times the number of students from the previous year.
- This means there is a 30% increase annually, but the best way to describe this increase in terms of a specific quantity is that each year the number of students is multiplied by 1.3.
Therefore, the statement "Each year, the number of students is [tex]\( \_\_\_\_\_\_\_\_ \)[/tex] the number the year before" can best be completed by:
A. 1.3 times
Thus, the correct answer is A. 1.3 times.
