Let's address the problem step-by-step to understand what happens when combining the functions [tex]\(A(t)\)[/tex] and [tex]\(B(t)\)[/tex].
The function for the number of credit cards opened at the first location is given by:
[tex]\[ A(t) = 10 + 2t \][/tex]
The function for the number of credit cards opened at the second location is:
[tex]\[ B(t) = 25 - t \][/tex]
### Combining the Two Functions
To find the total number of credit cards opened at both locations, we need to determine the expression for [tex]\( (A + B)(t) \)[/tex].
1. Sum of the two functions:
[tex]\[ (A + B)(t) = A(t) + B(t) \][/tex]
Substituting in the given functions:
[tex]\[ (A + B)(t) = (10 + 2t) + (25 - t) \][/tex]
2. Simplify the expression:
Combine like terms:
[tex]\[ (A + B)(t) = 10 + 2t + 25 - t \][/tex]
[tex]\[ (A + B)(t) = 35 + t \][/tex]
So, the expression to determine the total number of credit cards opened at the two locations is:
[tex]\[ (A + B)(t) = 35 + t \][/tex]
From the choices given:
- [tex]\((A+B)(t)=35+t\)[/tex] - This is correct.
- [tex]\((A+B)(t)=35+3t\)[/tex] - Incorrect, doesn't match our simplified expression.
### Difference of the Two Functions
To find the expression for the difference in the number of credit cards opened at both locations, we need to determine [tex]\( (A - B)(t) \)[/tex].
1. Difference of the two functions:
[tex]\[ (A - B)(t) = A(t) - B(t) \][/tex]
Substituting in the given functions:
[tex]\[ (A - B)(t) = (10 + 2t) - (25 - t) \][/tex]
2. Simplify the expression:
Distribute the negative sign and combine like terms:
[tex]\[ (A - B)(t) = 10 + 2t - 25 + t \][/tex]
[tex]\[ (A - B)(t) = -15 + 3t \][/tex]
So, the expression for the difference in the number of credit cards opened at the two locations is:
[tex]\[ (A - B)(t) = -15 + 3t \][/tex]
From the choices given:
- [tex]\((A-B)(t)=-15+t\)[/tex] - Incorrect.
- [tex]\((A-B)(t)=-15+3t\)[/tex] - This is correct.
### Summary
The correct expressions based on the given choices are:
[tex]\[ (A + B)(t) = 35 + t \][/tex]
[tex]\[ (A - B)(t) = -15 + 3t \][/tex]
