Question

Speeding Tickets In a certain state the maximum speed permitted on freeways is 65 mi/h, and the minimum is 40 mi/h. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum.

(a) Complete the expressions in the following piecewise defined function, where x is the speed at which you are driving.

$F\left(x\right)=\left\{\begin{array}{l}\underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}0<x<40\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}40\le x\le 65\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}x>65\end{array}\right\}$

(a) Find $F\left(30\right),F\left(50\right),\mathrm{and}F\left(75\right).$

(b) What do your answers in part (b) represent?

Short answer

(a) T function is defined as:$T\left(x\right)=\left\{\begin{array}{l}75x,\mathrm{if}\text{0}\le x\le 2\\ 50+50x,\mathrm{if}x>2\end{array}\right\}$

(b)

$\begin{array}{l}T\left(2\right)=150\\ T\left(3\right)=200\\ T\left(5\right)=300\end{array}$

(c)

The results represent the cost T of staying in the hotel for 2, 3 and 5 days.

Step-by-step solution

Part a. Step 1. Use the definition of functions.

A function is defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x, y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part a. Step 2. Analyze the information given.

We are given the function:

$F\left(x\right)=\left\{\begin{array}{l}\underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}0<x<40\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}40\le x\le 65\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}x>65\end{array}\right\}$

The maximum speed permitted on freeways is role="math" localid="1646138360047" $65\mathrm{mi}/\mathrm{h}$, and the minimum is role="math" localid="1646138459381" $40\mathrm{mi}/\mathrm{h}$. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum.

Part a. Step 3. Evaluate the function.

Thus if where x is the speed at which you are driving then

Between $40\mathrm{mi}/\mathrm{h}$to $65\mathrm{mi}/\mathrm{h}$there is no fine, for below $40\mathrm{mi}/\mathrm{h}$charge for x seed is $\mathrm{\$}15x$and for above $65\mathrm{mi}/\mathrm{h}$charge is $15\$\left(65-x\right)$.

Thus F function is defined as:

localid="1646139213828" $F\left(x\right)=\left\{\begin{array}{l}15x,\mathrm{if}0<x<40\\ 0,\mathrm{if}40\le x\le 65\\ 15\left(65-x\right),\mathrm{if}x>65\end{array}\right\}$

Part a. Step 4. Find values by plugging x-values.

.

Part b. Step 1. Use the definition of functions.

A function is defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x, y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part b. Step 2. Analyze the information given.

We are given the function:

$F\left(x\right)=\left\{\begin{array}{l}\underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}0<x<40\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}40\le x\le 65\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}x>65\end{array}\right\}$

The maximum speed permitted on freeways is $65\mathrm{mi}/\mathrm{h}$, and the minimum is $40\mathrm{mi}/\mathrm{h}$. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum.

Part b. Step 3. Find values by plugging x-values.

We are given the function:

$T\left(x\right)=\left\{\begin{array}{l}75x,\mathrm{if}\text{0}\le x\le 2\\ 50+50x,\mathrm{if}x>2\end{array}\right\}$

To find T values we proceed as follows:

$\begin{array}{l}T\left(x\right)=\left\{\begin{array}{l}75x,\mathrm{if}\text{0}\le x\le 2\\ 50+50x,\mathrm{if}x>2\end{array}\right\}\\ T\left(2\right)=75\left(2\right)=150\\ T\left(3\right)=50+50\left(3\right)=200\\ T\left(5\right)=50+50\left(5\right)=300\end{array}$

Part c. Step 1. Use the definition of functions.

A function is defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x, y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part c. Step 2. Analyze the information given.

We are given the function:

$F\left(x\right)=\left\{\begin{array}{l}\underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}0<x<40\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}40\le x\le 65\\ \underset{¯}{}\underset{¯}{}\underset{¯}{},\mathrm{if}x>65\end{array}\right\}$

The maximum speed permitted on freeways is $65\mathrm{mi}/\mathrm{h}$, and the minimum is $40\mathrm{mi}/\mathrm{h}$. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum.

Part c. Step 3. interpret the results.

$\begin{array}{l}T\left(2\right)=150\\ T\left(3\right)=200\\ T\left(5\right)=300\end{array}$

The results represent the cost T of staying in the hotel for 2, 3 and 5 days.