Sign In
| 14 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Mark always likes to have cans of his favorite soft drink in the refrigerator so that he can enjoy a cold drink whenever he wants.
At the beginning of the week, he had 11 cans in the fridge. Over the week, Mark wrote down each time cans were consumed or added.
A piecewise function is a function that is defined differently for different parts of its domain. These functions are commonly defined using intervals or inequalities. Consider the following example. f(x) = - x-2, & if x < 0 2x+1, & if x≥ 0 This function is defined as one linear function for values of x less than 0, and a different linear function for values of x that are at least zero. The graph of f is obtained by graphing the two rays. It should be noted that both rays represent the same function.
A piecewise function f is shown below. For every given value of x, find the correct value of f(x). Round the result to two decimal places, if necessary.
When graphing a piecewise function, each piece must be considered separately. First, a piece is graphed for the values of its domain. Then an end — or ends — of the piece are marked with one of the following.
The first piece is f(x) = 4x + 6. This is a linear function written in slope-intercept form. The function can be graphed by using the y-intercept and the slope.
Now, since the function is limited to inputs less than or equal to -1, the line will be graphed until it reaches x = -1. Since the inequality x≤ -1 is non-strict, the function is defined for x=-1 and the circle will be closed.
A similar process can be repeated to graph the second piece f(x) =- x+3. First, the line will be graphed.
This piece is defined for values of x greater than -1. This means that the line will be drawn starting at x=-1. Also, since the inequality is strict, the circle will be open.
Finally, the pieces will be graphed together in the same coordinate plane to complete the graph. It is important to pay attention to the limit points, since each value of x must be assigned to only one value of the function.
f(t) = 15t,& if 0≤ t ≤ 1 ? & ? & Since Heichi was biking at 15 miles per hour for one hour, he traveled 15 miles in this interval. In the next piece, the value of the function is constant at 15 for the values of t between 1 and 6 because he is just hanging out at the beach for these 5 hours. f(t) = 15t,& if 0≤ t ≤ 1 15, & if 1 < t ≤ 6 ? & Finally, when Heichi goes home, he bikes at 10 miles per hour. Since he is returning home, the distance from his house will decrease. Therefore, the distance traveled is written as the product of -10 and t. -10 t Also, since Heichi arrives at home one and a half hours later, the final value of the last piece should be 7.5 hours after the beginning of the trip. Therefore, the function needs to be translated 7.5 units right by subtracting 7.5 from t before multiplying by -10. - 10(t-7.5) = -10t + 75 Finally, it is possible to write the complete function rule by adding this final piece. f(t) = 15t,& if 0≤ t ≤ 1 15, & if 1 < t ≤ 6 -10t+75, & if6 < t ≤ 7.5
The second piece of the function is the horizontal line f(t) = 15. Also, since the inequality is non-strict, the circle at t=6 will be filled.
Now, the x-intercept can be used to graph the third and final piece. From Part A, it is known that the x-intercept is 7.5.
Finally, the graph of the piecewise function is completed.
Domain [0,7.5] The function's output represents how far Heichi is from his house. Looking at the graph, it can be seen that the minimum value of the function is 0, when Heichi is at his house, and the maximum value is 15, when Heichi is at the beach. While he is biking between his house and the beach, all distances from 0 to 15 are reached.
Range [0,15]A step function is a piecewise function that is defined by a constant value on each part of its domain. As an example, consider the following function. f(x) = 0, & if 0≤ x < 1 2, & if 1≤ x < 2 4, & if 2≤ x < 3 6, & if 3≤ x < 4 The graph of a step function consists of horizontal line segments, which can be interpreted as steps. The graph of the given function has four line segments.
As a summer activity, Tearrik participates in charity events for his community. He is volunteering for a food drive event this weekend.
He went to a shopping center multiple times over the week to collect boxes of food for the food drive. On each day, he collected the following number of boxes.
f(x) = 0, & if 0≤ x < 1 5, & if 1≤ x < 2 8, & if 2≤ x < 3 10, & if 3≤ x < 4 13, & if 4≤ x < 5 17, & if 5≤ x < 6
First, the domain of the function must be determined. The domain can be defined as the given days of the week, starting with the Sunday before Tearrik started collecting the boxes as x=0. Since Tearrik makes his first pickup on Monday, he starts with zero boxes on Sunday x=0. f(x) = 0, if 0≤ x < 1 On Monday, since one day has passed from Sunday, x=1. On this day, Tearrik collected his first 5 boxes. f(x) = 5, if 1≤ x < 2 On the following days, Tearrik collected more boxes. The boxes picked up each day were added to the number of boxes collected previously. Using this information, the pieces can be written in a table.
Day | Collected Boxes | Boxes in Total | x | f(x) |
---|---|---|---|---|
Sunday | - | 0 | 0≤ x < 1 | 0 |
Monday | 5 | 0 + 5 = 5 | 1≤ x < 2 | 5 |
Tuesday | 3 | 5 + 3 = 8 | 2≤ x < 3 | 8 |
Wednesday | 2 | 8 + 2 = 10 | 3≤ x < 4 | 10 |
Thursday | 3 | 10 + 3 = 13 | 4≤ x < 5 | 13 |
Friday | 4 | 13 + 4 = 17 | 5≤ x < 6 | 17 |
Finally, the function rule of the step function can be written based on the table. f(x) = 0, & if 0≤ x < 1 5, & if 1≤ x < 2 8, & if 2≤ x < 3 10, & if 3≤ x < 4 13, & if 4≤ x < 5 17, & if 5≤ x < 6
Since step functions are piecewise functions, to graph them, each piece must be considered separately. First, each piece is graphed in its domain as a part of a horizontal line. The end or ends are then marked with one of the following.
The first piece is defined over the interval 0≤ x < 2. This piece will be graphed by drawing a horizontal line at y=1 from x=0 to x=2.
It can be noted that x=0 is included in the domain of the first piece, but x=2 is not. Therefore, the left end of the segment will be marked with a closed circle and the right end with an open circle.
Then, the same process is repeated for each piece of the function.
Zain is working as a server in a restaurant for a week before their summer vacation ends.
Most of their payment comes from the tips they receive. Zain made note of how much they received in tips, starting from Monday and through their last day working on Saturday.
f(x) = 0, if0 ≤ x < 1 This indicates that Zain received $0 in tips on Sunday before they start working. On Monday x=1, Zain received $49.50 in tips. This defines the next piece of the function. f(x) = 49.50, if1≤ x < 2 Since the function reflects the total amount of money Zain receives in tips, the value of each of the pieces can be determined by adding the new tip value to the total tip amount from the day before. Using this information, the pieces can be written in a table.
Day | Tips Received | Total Tips | x | f(x) |
---|---|---|---|---|
Sunday | - | 0 | 0≤ x < 1 | 0 |
Monday | 49.50 | 0 + 49.50 = 49.50 | 1≤ x < 2 | 49.50 |
Tuesday | 38.17 | 49.50 + 38.17 = 87.67 | 2≤ x < 3 | 87.67 |
Wednesday | 41.45 | 87.67 + 41.45 = 129.12 | 3≤ x < 4 | 129.12 |
Thursday | 58.33 | 129.12 + 58.33 = 187.45 | 4≤ x < 5 | 187.45 |
Friday | 60.55 | 187.45 + 60.55 = 248.00 | 5≤ x < 6 | 248.00 |
Saturday | 57.00 | 248.00 + 57.00 = 305.00 | 6≤ x < 7 | 305.00 |
Finally, the step function can be written based on the table. f(x) = 0, &if 0≤ x < 1 49.50, &if 1 ≤ x < 2 87.67, &if 2 ≤ x < 3 129.12, &if 3 ≤ x < 4 187.45, &if 4 ≤ x < 5 248.00, &if 5 ≤ x < 6 305.00, &if 6 ≤ x < 7
The next piece is the horizontal segment at 49.50 with a closed endpoint at x=1 and an open endpoint at x=2.
The remaining pieces can be added to the graph by following the same process.
The greatest integer function, also known as the floor function, assigns the largest integer that is less than or equal to the value of x. This function is usually written as f(x)= ⌊ x ⌋ or f(x)= ||x||. Consider the following examples. ⌊ - 2 ⌋ & = - 2 ⌊ - 1.5 ⌋ & = - 2 ⌊ -0.01 ⌋ & = -1 ⌊ 1 ⌋ & = 1 ⌊ 1.25 ⌋ & = 1 It can be noted that if x is an integer, the function returns the same value. ⌊ - 2⌋ = - 2 ⌊ 0⌋ = 0 ⌊ 1 ⌋ = 1 Otherwise, it returns the closest integer at the left of x in a number line.
Considering more values can help understand how to draw the graph of the function.
x | f(x)=⌊ x ⌋ |
---|---|
- 1 | - 1 |
- 0.75 | - 1 |
- 0.5 | - 1 |
- 0.25 | - 1 |
0 | 0 |
0.5 | 0 |
1 | 1 |
1.25 | 1 |
1.75 | 1 |
1.99 | 1 |
2 | 2 |
From the table above, it can be seen that the function only changes its value when a new integer is reached. It can be noted that the greatest integer function is a step function. Its graph is presented as follows.
Dominika is going to a movie at a local theater on her last day of vacation.
The cost to park in the theater lot is $10 for less than an hour. An additional $2.50 is charged for each hour of parking.
f(x) = 2.5 ⌊ x ⌋ + 10
f_1(x) = 2.5 ⌊ x ⌋ Additionally, the lot requires an initial payment of $10 for the first 59 minutes of parking. This information can be used to add 10 to the value of the obtained function f_1. f(x) = 2.5 ⌊ x ⌋ + 10 This completes the required function, as the price to enter is considered and $2.50 is added as each hour is reached.
Next, since the greatest integer function is multiplied by 2.5, each value of y is multiplied by 2.5, vertically stretching the spaces between each horizontal segment.
Then, since 10 is added to the product, each segment is translated vertically 10 units up.
Finally, the scope of the coordinate plane will be adjusted so that more steps of the graph can be seen.
During his summer vacation, Ignacio went to private math lessons. After learning about the greatest integer function, Ignacio was asked by his math tutor to graph the numbers y greater than or equal to ⌊ x ⌋.
Graph the greatest integer function. Then, determine the solution set by shading the appropriate region.
The first step to graph an inequality is to graph the border function. The border function of the given inequality is given by the greatest integer function. This function is a step function whose output is the greatest integer less than or equal to the input x. Note that the inequality is non-strict, so the horizontal lines are drawn as solid lines.
Finally, the following is the complete graph of the inequality that the teacher asked for.
At the beginning of this lesson, it was asked that a function for the number of soda cans in Mark's refrigerator be written. Mark starts the week with 11 cans in the fridge and then does the following.
f(x) = 11, if 0 ≤ x < 1 Mark drank one can of soda on Monday. Therefore, he has now 11 - 1 = 10 cans left in the fridge that day. Also, since he neither drank nor bought any soda on Tuesday, the function's output is still 10 until Wednesday, when x=3. f(x) = 10, if1 ≤ x < 3 Mark bought 2 cans on Wednesday, meaning that there are 10 + 2 =12 cans of soda in the fridge that day. Since he neither bought nor drank any of the soda in the refrigerator on Thursday, he has 12 cans until Friday. f(x) = 12, if 3 ≤ x < 5 At Friday's party, a total of 10 cans were drunk. Therefore, he has 12 - 10 = 2 cans left in the refrigerator for Saturday. Then, on Saturday, he bought a 12-pack of soda cans, making it for 2 + 12 = 14 cans in the refrigerator for that day. Because no more information was given, the whole function rule can now be written. f(x) = 11, & if 0 ≤ x < 1 10, & if 1 ≤ x < 3 12, & if 3 ≤ x < 5 2, & if 5 ≤ x < 6 14, & if 6 ≤ x < 7
Mark is going to the beach on his last day of vacation and he plans to park his car in a parking lot near the beach. The parking lot charges $6 for the first 59 minutes and $3 for each hour of parking.
The first thing to notice is that $3 is charged for each hour of parking. This means that, as soon as one hour is reached, $3 is added to the total cost of parking. For example, parking a car for 1 hour and 10 minutes costs the same amount as parking the car for 1 hour and 55 minutes, since the next $3 increment will not be applied until the 2-hour mark is reached. Cost for1h 10min = Cost for 1h 55min The cost for either of the two noted time durations equals the rate per hour multiplied by the number of full hours parked — that is, 3* 1 = $3. In other words, at the one-hour mark, $3 was charged. In fact, if $3 is charged every hour, only the integer part of the number of hours determines the parking charges. Cost for1h 10min_(≈ 1.17 h) = Cost for 1h 55min_(≈ 1.92 h) This can be written algebraically by multiplying the greatest integer function ⌊ x ⌋ by 3, where x is the parking time in hours. 3⌊ x ⌋ However, the lot requires an initial payment of $6 for the first 59 minutes parked. Therefore, the total parking fee for both of the two example time intervals we mentioned before is $3+$6=$9. Therefore, we have to add 6 to the previous expression we wrote. C(x) = 3⌊ x ⌋ + 6 The previous function describes the cost of parking a car for x hours. Let's test it out!
Parking Time | Example Time | C(x) = 3⌊ x ⌋ + 6 | Cost |
---|---|---|---|
Less than one hour | 0:45=0.75 | C(0.75) = 3⌊ 0.75 ⌋ + 6 ⇓ C(0.75) = 3(0)+6=6 |
$6 |
Between 1 and 2 hours | 1:53=1.88 | C(1.88) = 3⌊ 1.88 ⌋ + 6 ⇓ C(1.88) = 3(1)+6=9 |
$9 |
Between 4 and 5 hours | 4:15=4.25 | C(4.25) = 3⌊ 4.25 ⌋ + 6 ⇓ C(4.25) = 3(4)+6=18 |
$18 |
As we can see, the total cost corresponds to the description of the parking lot rates. Let's graph this function on the coordinate plane.
From the function rule, we have that a= 3 and b= 6. Consequently, a* b = 3* 6=18.