Q(t) =

5, if 0 ¡Ü t ¡Ü 5
−t + 7 if 5 < t ¡Ü 8
t − 1
, if 8 < t ¡Ü 11

Q(x), 6 < x < 7 = ____
Q(m2 + 1), 7
< m ¡Ü 10
= 4

To find the value of Q(x) for the given interval, we need to look at the function defined by different ranges of t.

Given the function Q(t):
Q(t) = 5, if 0 ≤ t ≤ 5
Q(t) = -t + 7, if 5 < t ≤ 8
Q(t) = t - 1, if 8 < t ≤ 11

To find Q(x) for the interval 6 < x < 7, we need to determine the corresponding range for t. Since the function is not directly defined for x, we can replace it with t, and then solve for t.

From the given interval, we have:
6 < x < 7

To get the corresponding range for t, we can substitute x with t:
6 < t < 7

Now, we can determine the value of Q(t) for the range of t.

Looking at the function:
Q(t) = 5, if 0 ≤ t ≤ 5

Since the range of t (6 to 7) does not fall within this range, we need to check the next range.

Q(t) = -t + 7, if 5 < t ≤ 8

Since the range of t (6 to 7) falls within this range, we can substitute t with a value within this range to find the corresponding value of Q(t).

For simplicity, let's choose t = 6. Plugging t = 6 into the equation, we get:
Q(6) = -(6) + 7
= 1

Therefore, Q(x) for the interval 6 < x < 7 is 1.

For the second part of the question, we need to find Q(m^2 + 1) for the range of m from 7 to 10.

Using the given function, we can substitute t with m^2 + 1 in order to find Q(m^2 + 1).

Q(m^2 + 1) = 5, if 0 ≤ m^2 + 1 ≤ 5

The equation above gives us the range for m^2:
0 ≤ m^2 ≤ 4

Taking the square root of both sides, we get:
0 ≤ |m| ≤ 2

Since the range of m falls within this range (7 to 10), we can substitute m^2 + 1 with a value within this range to find the corresponding value of Q(m^2 + 1).

For simplicity, let's choose m = 7. Plugging m = 7 into the equation, we get:
Q(7^2 + 1) = Q(50) = t - 1
= 50 - 1
= 49

Therefore, Q(m^2 + 1) for the range 7 ≤ m ≤ 10 is 49.