Question

question 15 (multiple choice worth 1 points)
(exponential functions and function combinations mc)

a retail clothing store offers customers an opportunity to open up a credit card during checkout. one location of the retail clothing store states that the number of credit cards, \\(a\\), that are opened \\(t\\) months since january can be modeled by the function \\(a(t) = 10 + 2t\\). the number of credit cards opened at another location, \\(b\\), is defined by the function \\(b(t) = 25 - t\\). what is an expression that can be used to determine the total amount of credit cards opened at the two locations?

\\((a + b)(t) = 35 + t\\)

\\((a + b)(t) = 35 + 3t\\)

\\((a - b)(t) = -15 + t\\)

\\((a - b)(t) = -15 + 3t\\)

Explanation:

Identify given functions

The problem provides two functions representing the number of credit cards opened at two different store locations, \(A\) and \(B\), as a function of time \(t\) (in months):

$$A(t) = 10 + 2t$$

$$B(t) = 25 - t$$

Determine the required operation

To find the total amount of credit cards opened at both locations, we need to add the two functions together. This is represented by the function combination:

$$(A + B)(t) = A(t) + B(t)$$

Perform function addition

Using the Function Evaluation concept to substitute the expressions:

$$(A + B)(t) = (10 + 2t) + (25 - t)$$

Combine the constant terms and the variable terms:

$$(A + B)(t) = (10 + 25) + (2t - t)$$

$$(A + B)(t) = 35 + t$$

Answer:

• (A) \((A + B)(t) = 35 + t\) (Correct answer)
• (B) \((A + B)(t) = 35 + 3t\)
• (C) \((A - B)(t) = -15 + t\)
• (D) \((A - B)(t) = -15 + 3t\)