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| 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 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.
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.
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)=⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧0,5,8,10,13,17,if 0≤x<1if 1≤x<2if 2≤x<3if 3≤x<4if 4≤x<5if 5≤x<6
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 |
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.
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 |
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.
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
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.
Before evaluating the function at x=- 43, we first need to identify which interval of the domain of the function - 43 belongs to. Since - 43 is about -1.33, it lies between -3 and 0. As such, the given input belongs to the second interval. f(x) = 2x-3, &if x ≤ -3 -3x, &if -3 < x ≤ 0 6, &if 0 < x ≤ 2 x+6, &if x > 2 Therefore, to evaluate the function at - 43 we will use the second expression, -3x.
As in the previous part, let's first determine the interval of the domain that includes the input 2.1. Because 2.1 > 2, it belongs to the last interval. f(x) = 2x-3, &if x ≤ -3 -3x, &if -3 < x ≤ 0 6, &if 0 < x ≤ 2 x+6, &if x > 2 Therefore, this time we will use the last expression of f(x), x+6.
We begin by noting that the given expression involves four different inputs, namely, -3, 4, 2, and -1. Therefore, to know which expression of the function we need to use for evaluating each input, let's locate them in the corresponding domain's intervals.
Having identified the expression we need to use for each input, we can substitute the inputs into their expressions and find the corresponding values. To do so, let's use a table of values!
Input | Interval | Function | Evaluation |
---|---|---|---|
-3 | x ≤ -3 | f(x) = 2x-3 | f(-3) = 2(-3)-3 = -9 |
4 | x > 2 | f(x) = x+6 | f(4) = 4+6 = 10 |
2 | 0 < x ≤ 2 | f(x) = 6 | f(2) = 6 |
-1 | -3 < x ≤ 0 | f(x) = -3x | f(-1) = -3(-1) = 3 |
Finally, let's substitute these values into the given expression and simplify.
The following graph represents the cost, in dollars, of buying x tickets to a magic show.
To determine the cost of buying 14 tickets, we need to determine the y-coordinate of the point for which x=14. To do so, let's draw the line x=14. Keep in mind that the point must lie on the graph of the given piecewise function.
We see that the line intersects the graph at the point (14,125). Therefore, buying 14 tickets has a cost of $125.
To determine whether it is cheaper to buy 4 or 5 tickets, let's start by finding the cost of buying 4 tickets. To do this, let's draw the line x=4.
Note that the point (4,50) belongs to the graph, while the point filled with white does not. Therefore, the cost of buying 4 tickets is $50. Now, let's draw the line x=5. It will be halfway between 4 and 6.
It can be seen that the line x=5 intersects the graph at a point whose y-coordinate is less than 50. This means that buying 5 tickets costs less than $50. Consequently, it is cheaper to buy 5 tickets than to buy 4 tickets.
To determine the maximum number of tickets that we can buy with $125, let's draw the line y=125.
We can see the line intersects the graph at two different points, (14,125) and (18,125). From this we can conclude that buying 14 tickets costs the same amount of money as buying 18 tickets. cc Number of Tickets & Cost [0.15cm] 14 & $125 18 & $125 Consequently, we can buy a maximum of 18 tickets with $125. In fact, note that we can also buy 17 tickets with $125 but we cannot buy 15 or 16 tickets for this price!
From the definition of f(x), we can see that the domain covers the entire number line — that is, the domain of the function is all the real numbers.
As we can see, the graph of f(x) does not have any horizontal gaps between the pieces. Knowing this, we can discard option D because it has a gap between x=3 and x=4.
On the other hand, note that the third domain of the function includes both endpoints, 0≤ x ≤ 3. Therefore, third piece of graph, from left to right, has to include both endpoints. Looking at the remaining three graphs, we can see that option A does not fulfill this condition, so we can discard option A.
As we can see, graphs B and C both have the correct distribution of the endpoints. Let's now take a look at each branch to check if they correspond to the function rule defined for each interval. For simplicity, let's start with the second piece of f(x), f(x)=-2.5. Note that in graph B, the second piece is graphed at y=-4. Therefore, we will also discard graph B.
Finally, let's verify that each part of graph C corresponds to the definition of f(x).
Consequently, the correct option is C.
First, notice that both parts of g(x) involve the greatest integer function y=⌊ x ⌋, which is a step function. As such, both pieces of the graph of g(x) should look like a step function, similar to stairs. This leads us to discard graph C.
Note that the left-hand piece of graph A looks like a step function but the right-hand piece does not. Therefore, we can also discard graph A.
In contrast, graphs B and D both look like a step function on both branches of their graphs. Therefore, we need to make a deeper analysis. Note that both graphs are exactly the same for x≥ 0, which corresponds to the graph of y=⌊ x ⌋. However, for x< 0, the endpoints of some segments are different. Let's point out all the differences.
To determine which is the correct graph, we will test a point. For example, let's find g(-2). Since -2 < 0, we have to use the first part of g(x) to evaluate it. g(x) = -⌊ x ⌋ ⇓ g( -2) = -⌊ -2 ⌋ Since -2 is an integer, ⌊ -2 ⌋=-2. Let's substitute it into the right-hand side. g(-2) = -(-2) = 2 Consequently, the point (-2,2) belongs to the graph of g(x). We can see that graph B fulfills this condition, but graph D does not. The corresponding point on graph D is (-2,3). Therefore, we can discard graph D and conclude that the correct option is graph B.
Note that the left-hand piece is the graph of y=⌊ x ⌋ but, because of the minus sign in front of the function rule of the first interval, it is reflected in the x-axis.