Answer :
To determine the positive and negative intervals of the function [tex]\( r(x) = 3 \cdot (0.5)^x - 6 \)[/tex], we can analyze the function's output at various values of [tex]\( x \)[/tex] and identify the regions where the function is greater than zero (positive intervals) and where it is less than zero (negative intervals).
### Step-by-Step Analysis
1. Evaluate the function at various points:
- For [tex]\( x = -10 \)[/tex]:
[tex]\[ r(-10) = 3 \cdot (0.5)^{-10} - 6 = 3 \cdot 1024 - 6 = 3066 \][/tex]
(positive)
- For [tex]\( x = -9 \)[/tex]:
[tex]\[ r(-9) = 3 \cdot (0.5)^{-9} - 6 = 3 \cdot 512 - 6 = 1530 \][/tex]
(positive)
- For [tex]\( x = -8 \)[/tex]:
[tex]\[ r(-8) = 3 \cdot (0.5)^{-8} - 6 = 3 \cdot 256 - 6 = 762 \][/tex]
(positive)
- For [tex]\( x = -7 \)[/tex]:
[tex]\[ r(-7) = 3 \cdot (0.5)^{-7} - 6 = 3 \cdot 128 - 6 = 378 \][/tex]
(positive)
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ r(-6) = 3 \cdot (0.5)^{-6} - 6 = 3 \cdot 64 - 6 = 186 \][/tex]
(positive)
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ r(-5) = 3 \cdot (0.5)^{-5} - 6 = 3 \cdot 32 - 6 = 90 \][/tex]
(positive)
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ r(-4) = 3 \cdot (0.5)^{-4} - 6 = 3 \cdot 16 - 6 = 42 \][/tex]
(positive)
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ r(-3) = 3 \cdot (0.5)^{-3} - 6 = 3 \cdot 8 - 6 = 18 \][/tex]
(positive)
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ r(-2) = 3 \cdot (0.5)^{-2} - 6 = 3 \cdot 4 - 6 = 6 \][/tex]
(positive)
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ r(-1) = 3 \cdot (0.5)^{-1} - 6 = 3 \cdot 2 - 6 = 0 \][/tex]
(zero)
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ r(0) = 3 \cdot (0.5)^0 - 6 = 3 \cdot 1 - 6 = -3 \][/tex]
(negative)
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ r(1) = 3 \cdot (0.5)^1 - 6 = 3 \cdot 0.5 - 6 = -4.5 \][/tex]
(negative)
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ r(2) = 3 \cdot (0.5)^2 - 6 = 3 \cdot 0.25 - 6 = -5.25 \][/tex]
(negative)
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ r(3) = 3 \cdot (0.5)^3 - 6 = 3 \cdot 0.125 - 6 = -5.625 \][/tex]
(negative)
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ r(4) = 3 \cdot (0.5)^4 - 6 = 3 \cdot 0.0625 - 6 = -5.8125 \][/tex]
(negative)
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ r(5) = 3 \cdot (0.5)^5 - 6 = 3 \cdot 0.03125 - 6 = -5.90625 \][/tex]
(negative)
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ r(6) = 3 \cdot (0.5)^6 - 6 = 3 \cdot 0.015625 - 6 = -5.953125 \][/tex]
(negative)
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ r(7) = 3 \cdot (0.5)^7 - 6 = 3 \cdot 0.0078125 - 6 = -5.9765625 \][/tex]
(negative)
- For [tex]\( x = 8 \)[/tex]:
[tex]\[ r(8) = 3 \cdot (0.5)^8 - 6 = 3 \cdot 0.00390625 - 6 = -5.98828125 \][/tex]
(negative)
- For [tex]\( x = 9 \)[/tex]:
[tex]\[ r(9) = 3 \cdot (0.5)^9 - 6 = 3 \cdot 0.001953125 - 6 = -5.994140625 \][/tex]
(negative)
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ r(10) = 3 \cdot (0.5)^{10} - 6 = 3 \cdot 0.0009765625 - 6 = -5.9970703125 \][/tex]
(negative)
2. Identify the sign of the function:
- Positive Intervals: When [tex]\( x = -10, -9, -8, -7, -6, -5, -4, -3, -2 \)[/tex]. For these values, the function [tex]\( r(x) \)[/tex] outputs positive numbers.
- Negative Intervals: When [tex]\( x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \)[/tex]. For these values, the function [tex]\( r(x) \)[/tex] outputs negative numbers.
3. Summarize the intervals:
- [tex]\( r(x) > 0 \)[/tex] for [tex]\( x \in \{ -10, -9, -8, -7, -6, -5, -4, -3, -2 \} \)[/tex]
- [tex]\( r(x) < 0 \)[/tex] for [tex]\( x \in \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \)[/tex]
Thus, the positive interval for the function [tex]\( r(x) = 3 \cdot (0.5)^x - 6 \)[/tex] is when [tex]\( x \)[/tex] values are from [tex]\(-10\)[/tex] to [tex]\(-2\)[/tex]. The negative interval is when [tex]\( x \)[/tex] values are from [tex]\( 0 \)[/tex] to [tex]\( 10 \)[/tex].
### Step-by-Step Analysis
1. Evaluate the function at various points:
- For [tex]\( x = -10 \)[/tex]:
[tex]\[ r(-10) = 3 \cdot (0.5)^{-10} - 6 = 3 \cdot 1024 - 6 = 3066 \][/tex]
(positive)
- For [tex]\( x = -9 \)[/tex]:
[tex]\[ r(-9) = 3 \cdot (0.5)^{-9} - 6 = 3 \cdot 512 - 6 = 1530 \][/tex]
(positive)
- For [tex]\( x = -8 \)[/tex]:
[tex]\[ r(-8) = 3 \cdot (0.5)^{-8} - 6 = 3 \cdot 256 - 6 = 762 \][/tex]
(positive)
- For [tex]\( x = -7 \)[/tex]:
[tex]\[ r(-7) = 3 \cdot (0.5)^{-7} - 6 = 3 \cdot 128 - 6 = 378 \][/tex]
(positive)
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ r(-6) = 3 \cdot (0.5)^{-6} - 6 = 3 \cdot 64 - 6 = 186 \][/tex]
(positive)
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ r(-5) = 3 \cdot (0.5)^{-5} - 6 = 3 \cdot 32 - 6 = 90 \][/tex]
(positive)
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ r(-4) = 3 \cdot (0.5)^{-4} - 6 = 3 \cdot 16 - 6 = 42 \][/tex]
(positive)
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ r(-3) = 3 \cdot (0.5)^{-3} - 6 = 3 \cdot 8 - 6 = 18 \][/tex]
(positive)
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ r(-2) = 3 \cdot (0.5)^{-2} - 6 = 3 \cdot 4 - 6 = 6 \][/tex]
(positive)
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ r(-1) = 3 \cdot (0.5)^{-1} - 6 = 3 \cdot 2 - 6 = 0 \][/tex]
(zero)
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ r(0) = 3 \cdot (0.5)^0 - 6 = 3 \cdot 1 - 6 = -3 \][/tex]
(negative)
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ r(1) = 3 \cdot (0.5)^1 - 6 = 3 \cdot 0.5 - 6 = -4.5 \][/tex]
(negative)
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ r(2) = 3 \cdot (0.5)^2 - 6 = 3 \cdot 0.25 - 6 = -5.25 \][/tex]
(negative)
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ r(3) = 3 \cdot (0.5)^3 - 6 = 3 \cdot 0.125 - 6 = -5.625 \][/tex]
(negative)
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ r(4) = 3 \cdot (0.5)^4 - 6 = 3 \cdot 0.0625 - 6 = -5.8125 \][/tex]
(negative)
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ r(5) = 3 \cdot (0.5)^5 - 6 = 3 \cdot 0.03125 - 6 = -5.90625 \][/tex]
(negative)
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ r(6) = 3 \cdot (0.5)^6 - 6 = 3 \cdot 0.015625 - 6 = -5.953125 \][/tex]
(negative)
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ r(7) = 3 \cdot (0.5)^7 - 6 = 3 \cdot 0.0078125 - 6 = -5.9765625 \][/tex]
(negative)
- For [tex]\( x = 8 \)[/tex]:
[tex]\[ r(8) = 3 \cdot (0.5)^8 - 6 = 3 \cdot 0.00390625 - 6 = -5.98828125 \][/tex]
(negative)
- For [tex]\( x = 9 \)[/tex]:
[tex]\[ r(9) = 3 \cdot (0.5)^9 - 6 = 3 \cdot 0.001953125 - 6 = -5.994140625 \][/tex]
(negative)
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ r(10) = 3 \cdot (0.5)^{10} - 6 = 3 \cdot 0.0009765625 - 6 = -5.9970703125 \][/tex]
(negative)
2. Identify the sign of the function:
- Positive Intervals: When [tex]\( x = -10, -9, -8, -7, -6, -5, -4, -3, -2 \)[/tex]. For these values, the function [tex]\( r(x) \)[/tex] outputs positive numbers.
- Negative Intervals: When [tex]\( x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \)[/tex]. For these values, the function [tex]\( r(x) \)[/tex] outputs negative numbers.
3. Summarize the intervals:
- [tex]\( r(x) > 0 \)[/tex] for [tex]\( x \in \{ -10, -9, -8, -7, -6, -5, -4, -3, -2 \} \)[/tex]
- [tex]\( r(x) < 0 \)[/tex] for [tex]\( x \in \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \)[/tex]
Thus, the positive interval for the function [tex]\( r(x) = 3 \cdot (0.5)^x - 6 \)[/tex] is when [tex]\( x \)[/tex] values are from [tex]\(-10\)[/tex] to [tex]\(-2\)[/tex]. The negative interval is when [tex]\( x \)[/tex] values are from [tex]\( 0 \)[/tex] to [tex]\( 10 \)[/tex].
