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
Dominika plans to go to the waterfall this afternoon after walking through the woods. The path that she will follow is a bit bumpy and, after the wood end, it consists of one hill with a flat area before and after it. Because of this path, her speed will change along the way, but Dominika knows she will reach the waterfall in half an hour.
Dominika left the house at 3:47PM, but before she left, she wrote the following function on the refrigerator door so that her mother would how many meters from home Dominika will be at any given moment of the walk. d(t) = 81t, &if 0 ≤ t < 7 60t + 147, &if 7 ≤ t ≤ 15 87t - 258, &if 15 < t < 23 81t - 120, &if 23 ≤ t ≤ 30 Here, t represents the minutes since Dominika left home.
At 4:00PM, 13 minutes have passed since Dominika left home. Therefore, to know how far away Dominika is, we need to evaluate d(t) at t=13. Since 13 is between 7 and 15, we need to use the second piece of the function. d(t) = 81t, &if 0 ≤ t < 7 60t + 147, &if 7 ≤ t ≤ 15 87t - 258, &if 15 < t < 23 81t - 120, &if 23 ≤ t ≤ 30 We are ready to find out how far from home Dominika was at 4:00PM.
To determine how far away the waterfall is, we need to know how long it takes Dominika to get there. However, rememeber that Dominika said she would reach the waterfall in half an hour. Therefore, we need to evaluate d(t) at t=30 minutes. To do this, we will use the last part of the function. d(t) = 81t, &if 0 ≤ t < 7 60t + 147, &if 7 ≤ t ≤ 15 87t - 258, &if 15 < t < 23 81t - 120, &if 23 ≤ t ≤ 30 Let's find out how far from Dominika's house the waterfall is.
Despite we are not specified how the path is, we do know it consists of one hill preceded and followed by flat regions. Therefore, Dominika should walk at three different speeds.
Consequently, these three speeds should appear in the function rule. To identify them, note that d(t) is formed by four linear functions, and the slope of a linear function can be interpreted as the speed of a moving object. Therefore, let's identify Dominika's speed at each time interval in a table.
Function | Interval | Slope or Speed (m/min) |
---|---|---|
81t | 0 ≤ t < 7 | 81 |
60t+147 | 7 ≤ t ≤ 15 | 60 |
87t-258 | 15 < t < 23 | 87 |
81t-120 | 23 ≤ t ≤ 30 | 81 |
Taking into account that walking uphill slows the speed down, and using the information in the table, let's pair each speed to the corresponding part of the path. cccl Speed & & Path & [0.15cm] 60 m/min & & Uphill &↗ 81 m/min & & Flat & → 87 m/min & & Downhill & ↘ Consequently, Dominika walked uphill during the second time interval. That is, she walked uphill for 15-7 = 8 minutes.
Let f(x) be a piecewise function that consists of two linear functions. The graph of the function is shown.
We start by noticing that the graph of f(x) consists of two pieces. The left-hand piece is defined for x≤ 1 and the right-hand piece is defined for x >1.
Since 16>1, we need to use the right-hand piece to evaluate f(16). However, the value for x=16 is not shown in the graph and we do not know the function rule. Therefore, let's first find the equation of the right-hand piece. To do so, we can use the point-slope form equation. y-y_1 = m(x-x_1) Let's highlight the slope and one point of the right-hand piece.
Now we are ready to find the equation of the right-hand piece.
We have found the equation of the right-hand piece. Now we can proceed with evaluating f(x) at x=16.
Now we have to evaluate f(x) at x=-5. Since -5 < 1, we have to use the left-hand piece. Just like in Part A, we do not know its equation, but we can find it in a similar fashion. However, this time we will use the slope-intercept form equation.
The slope of the line is -2 and the y-intercept is -1. We can use this information to write the equation of the line. y = mx+b ⇓ y = -2x-1 Using this equation and the one we found in Part A, we can write the complete function rule defining f(x). f(x) = -2x-1, &ifx≤ 1 12x + 32, &ifx>1 Finally, let's evaluate f(x) at x=-5.
We are given two piecewise functions, each involving the greatest integer function. Recall that the floor function ⌊ x ⌋ assigns the largest integer that is less than or equal to the value of x. Let's start by identifying the four numbers at which we have to evaluate f(x) and g(x). f( ↑-0.75) + f( ↑2.98)* g( ↑7) - g( ↑-3.05) Since -0.75 is less than 1, we have to use the first piece of f(x) to evaluate it.
Next, let's find f( 2.98). Since 2.98> 1, we will use the second part of f(x).
Now, let's evaluate g(x) at x= 7. Since 7 is greater than -2, we have to use the second part of g(x).
Last but not least, we will find g(-3.05). Here, we will use the first part of g(x) because -3.05 is less than -2.
Finally, let's substitute the values we found into the given expression and simplify.