a band must have the right mix of instruments. use the graph of the step function to find how many tubas…

Explanation:

Step1: Analyze the step function graph

The x - axis represents the number of trumpet players and the y - axis represents the number of tuba players. We need to find the value of y (tuba players) when x (trumpet players) = 27.

Step2: Locate x = 27 on the x - axis

Looking at the step function graph, we check the intervals. For the interval where x is in [24, 28), the y - value (tuba players) is 7? Wait, no, wait. Wait, let's check the graph again. Wait, the points: when x is from 0 (open circle) to 4 (closed circle), y = 1; from 4 (open) to 8 (closed), y = 2; 8 (open) to 12 (closed), y = 3; 12 (open) to 16 (closed), y = 4; 16 (open) to 20 (closed), y = 5; 20 (open) to 24 (closed), y = 6; 24 (open) to 28 (closed), y = 7; 28 (open) to 32 (closed), y = 8. Wait, 27 is in the interval [24, 28) (since 24 < 27 ≤ 28? Wait, no, the open circle at 24, closed at 28? Wait, the graph: at x = 24, open circle, then the step goes to y = 7, and at x = 28, closed circle? Wait, no, looking at the graph: the steps are:

• From x = 0 (open) to x = 4 (closed): y = 1

• x = 4 (open) to x = 8 (closed): y = 2

• x = 8 (open) to x = 12 (closed): y = 3

• x = 12 (open) to x = 16 (closed): y = 4

• x = 16 (open) to x = 20 (closed): y = 5

• x = 20 (open) to x = 24 (closed): y = 6

• x = 24 (open) to x = 28 (closed): y = 7

• x = 28 (open) to x = 32 (closed): y = 8

Wait, 27 is between 24 (open) and 28 (closed), so the value of the step function at x = 27 is 7? Wait, no, wait the closed circle at 28 is y = 8? Wait, no, the graph has: at x = 24, open circle (y = 7), then the line goes to x = 28, closed circle (y = 7)? Wait, no, the points: the open circle at x = 24 is y = 7? Wait, no, let's look at the y - axis labels. The y - axis is tuba players, with 1,2,3,4,5,6,7,8. The steps:

• First step: from x = 0 (open) to x = 4 (closed), y = 1 (closed circle at x = 4, y = 1)

• Second step: x = 4 (open) to x = 8 (closed), y = 2 (closed at x = 8, y = 2)

• Third: x = 8 (open) to x = 12 (closed), y = 3 (closed at x = 12, y = 3)

• Fourth: x = 12 (open) to x = 16 (closed), y = 4 (closed at x = 16, y = 4)

• Fifth: x = 16 (open) to x = 20 (closed), y = 5 (closed at x = 20, y = 5)

• Sixth: x = 20 (open) to x = 24 (closed), y = 6 (closed at x = 24, y = 6)

• Seventh: x = 24 (open) to x = 28 (closed), y = 7 (closed at x = 28, y = 7? Wait, no, the next step: open circle at x = 28, y = 8, and closed circle at x = 32, y = 8. Wait, I think I misread. Let's list the intervals with their y - values:

• Interval (0, 4]: y = 1 (closed at 4, open at 0)

• (4, 8]: y = 2 (closed at 8, open at 4)

• (8, 12]: y = 3 (closed at 12, open at 8)

• (12, 16]: y = 4 (closed at 16, open at 12)

• (16, 20]: y = 5 (closed at 20, open at 16)

• (20, 24]: y = 6 (closed at 24, open at 20)

• (24, 28]: y = 7 (closed at 28, open at 24)

• (28, 32]: y = 8 (closed at 32, open at 28)

Now, x = 27 is in the interval (24, 28], so the value of the step function (number of tuba players) is 7? Wait, no, wait the closed circle at x = 28: what's the y - value? Looking at the graph, the closed circle at x = 28 is y = 7? Wait, no, the open circle at x = 28 is y = 8, and the closed circle at x = 32 is y = 8. Wait, the steps:

• At x = 4: closed circle, y = 1

• x = 8: closed, y = 2

• x = 12: closed, y = 3

• x = 16: closed, y = 4

• x = 20: closed, y = 5

• x = 24: closed, y = 6

• x = 28: closed, y = 7

• x = 32: closed, y = 8

And open circles at x = 0,4,8,12,16,20,24,28 with y - values 1,2,3,4,5,6,7,8 respectively.

So for x = 27, which is between 24 (closed, y = 6) and 28 (closed, y = 7)? Wait, no, the interval for y = 7 is (24, 28], because the open circle at x = 24 is y = 7, and closed at x = 28 is y = 7? Wait, I'm getting confused. Let's use the definition of a step function: the value at a point is the value of the step that contains the point, with closed circles included and open circles excluded.

So:

• For x in (0, 4], y = 1 (since x = 4 is closed, included; x = 0 is open, excluded)

• x in (4, 8], y = 2

• x in (8, 12], y = 3

• x in (12, 16], y = 4

• x in (16, 20], y = 5

• x in (20, 24], y = 6

• x in (24, 28], y = 7

• x in (28, 32], y = 8

Now, 27 is in (24, 28], so y = 7? Wait, no, wait x = 24 is closed, y = 6. Then x = 24 < x ≤ 28: what's the y - value? The open circle at x = 24 is y = 7, and the closed circle at x = 28 is y = 7? Wait, maybe the graph is such that when x is 24 or more but less than 28, the number of tubas is 7? Wait, let's check the x - value 27. So 27 is greater than 24 and less than 28. So we look at the step where x is in [24, 28) (open at 28) or (24, 28] (closed at 28). From the graph, the closed circle at x = 28 is y = 7? Wait, the graph shows: at x = 24, open circle (y = 7), then a horizontal line to x = 28, closed circle (y = 7), then open circle at x = 28 (y = 8) and horizontal line to x = 32, closed circle (y = 8). So for x = 27, which is between 24 and 28, the y - value (tuba players) is 7? Wait, no, wait the y - axis: when x = 24, the closed circle is y = 6, open circle y = 7. Then from x = 24 (open) to x = 28 (closed), y = 7. Then x = 28 (open) to x = 32 (closed), y = 8. So 27 is in (24, 28], so y = 7? Wait, no, 24 < 27 ≤ 28, so the step for that interval is y = 7. So when there are 27 trumpets, the number of tubas is 7? Wait, no, wait let's count the steps again. Let's list the x - values and corresponding y - values:

• 0 < x ≤ 4: y = 1

• 4 < x ≤ 8: y = 2

• 8 < x ≤ 12: y = 3

• 12 < x ≤ 16: y = 4

• 16 < x ≤ 20: y = 5

• 20 < x ≤ 24: y = 6

• 24 < x ≤ 28: y = 7

• 28 < x ≤ 32: y = 8

Yes, that makes sense. So 27 is in (24, 28], so y = 7. Wait, but wait, the closed circle at x = 24 is y = 6, which is for x ≤ 24. Then x > 24 and ≤ 28: y = 7. So 27 is in that interval, so tubas = 7? Wait, no, maybe I made a mistake. Wait, let's check the graph again. The vertical axis is tuba players, horizontal is trumpet players. The steps:

• First step: from x = 0 (open) to x = 4 (closed), y = 1 (so when you have 4 trumpets, tubas = 1)

• Second: x = 4 (open) to x = 8 (closed), y = 2 (8 trumpets, tubas = 2)

• Third: x = 8 (open) to x = 12 (closed), y = 3 (12 trumpets, tubas = 3)

• Fourth: x = 12 (open) to x = 16 (closed), y = 4 (16 trumpets, tubas = 4)

• Fifth: x = 16 (open) to x = 20 (closed), y = 5 (20 trumpets, tubas = 5)

• Sixth: x = 20 (open) to x = 24 (closed), y = 6 (24 trumpets, tubas = 6)

• Seventh: x = 24 (open) to x = 28 (closed), y = 7 (28 trumpets, tubas = 7)

• Eighth: x = 28 (open) to x = 32 (closed), y = 8 (32 trumpets, tubas = 8)

So for 27 trumpets, which is between 24 and 28 (24 < 27 ≤ 28), the number of tubas is 7? Wait, no, 24 < 27 < 28, so it's in the interval (24, 28), and the step function at x = 27 is 7. So the number of tubas needed is 7? Wait, but let's check the graph again. The closed circle at x = 28 is y = 7, and open circle at x = 28 is y = 8. So when x = 27, which is less than 28, the y - value is 7. So the answer is 7? Wait, no, wait maybe I messed up the intervals. Let's see: the x - axis is trumpet players, y - axis tuba players. The step function:

• For 0 < trumpets ≤ 4: tubas = 1

• 4 < trumpets ≤ 8: tubas = 2

• 8 < trumpets ≤ 12: tubas = 3

• 12 < trumpets ≤ 16: tubas = 4

• 16 < trumpets ≤ 20: tubas = 5

• 20 < trumpets ≤ 24: tubas = 6

• 24 < trumpets ≤ 28: tubas = 7

• 28 < trumpets ≤ 32: tubas = 8

So 27 is in 24 < 27 ≤ 28, so tubas = 7. Wait, but the closed circle at x = 28 is y = 7, so when trumpets = 28, tubas = 7, and when trumpets > 28 (up to 32), tubas = 8. So for 27 trumpets, tubas = 7.

Answer:

7