Functions and Function Notation

We are told that the hourly rate in the parking lot is $2.50 and that a flat fee is also charged. Price per hour & → $2.50 Flat fee & → $0.75 Based on this information, the cost for parking a car equals the product of the hourly rate and the number of hours parked plus the flat fee. Let's translate this phrase into an algebraic expression. C(t) = 2.5 t + 0.75 This function represents the cost of parking a car for t hours.

Let's start by writing the function we found that models the cost of parking a car for t hours. C(t) = 2.5t + 0.75 We know Kevin parked his car for four and a half hours. To determine how much will he pay when he leaves, we will evaluate C(t) at t=4.5.

Kevin will pay $12 for parking the car for four and a half hours.

The fact that Jordan paid $14.50 means that the function C(t) is equal to 14.50. C(t) = 14.50 We want to find the value of t that produced 14.50 as the output. To find it, we will substitute 14.50 for C(t) into the function rule found in the first part and solve the resulting equation for t.

We found that the output of C(t) is 14.50 when the input is 5.5. This means that Jordan parked her car for 5.5 hours.

Applying the gift card first is the same as applying G to the displayed price, so we will start by applying G to x. x → G(x) Next, Dylan will apply the store's discount. This means that he will apply the function S to the output of G. x → G(x) → S( G(x) ) Therefore, the composition of S of G represents the cost that Dylan will pay, based on his decision. That is, f(x)=S(G(x)). To find this composition, we will substitute G(x) for x in the function rule of S(x).

Finally, to find the explicit rule of f(x), we will substitute x-10 for G(x).

In contrast to the previous part, now Dylan wants to apply the discounts in the opposite order. That is, he wants to apply the store's discount first. Therefore, the function S will be applied to the displayed price. x → S(x) Then Dylan will apply the gift card. This means that he will apply the function G to the output of S. x → S(x) → G(S(x)) The composition of G of S represents the cost that Dylan will pay, based on the order he picked. That is, g(x)=G(S(x)). As before, to find this composition, we will start by writing the function rule of G(x) and then substitute S(x) for x.

To determine the order that gives the maximum benefit to Dylan, we will find f(24) and g(24) and compare them.

This means that by applying the gift card first and the store's discount second, Dylan will pay $11.90 for the shoes. Let's now find g(24).

From this, we can say that by applying the store's discount first and the gift card second, Dylan will pay $10.40 for the shoes, which is $1.50 less than when using the opposite discount order. As such, the answer is option B. Store's discount first, then gift card ✓

To determine the function rule that defines f(x), we need to find the values of a and b involved in the general form of f(x). f(x) = ax + b We can find these values by using the given equation. f(f(x) ) = 25x+42 The left-hand side of the equation implies that we need to perform the composition of f(x) with itself. To do this, in the general function rule, we substitute f(x) for x.

We can now substitute ax+b for f(x) on the right-hand side.

The resulting expression on the right-hand side should be equal to 25x + 42. For them to be equal, the coefficients of the x-terms and the constant term must be the same on both sides. Let's highlight them. a^2x + ab+b = 25x+42 From this equation, we see that a^2 must equal 25 and that ab+b must equal 42. Therefore, there are two possible values for a, 5 and -5. 5^2 &= 25 [1ex] ( -5)^2 &= 25 However, we are told that a is positive, which leads us to discard the negative option. Therefore, a=5. With this information, let's find b.

Since we already know the values of a and b, we can write the function rule that defines f(x). f(x) = ax+b ⇒ f(x) = 5x + 7