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Sagot :
To find the lattice points of the inverse function, we need to take the inverse of the function [tex]\( f(x) = -\log_3(x + 3) + 9 \)[/tex]. Follow these steps to find the inverse:
1. Set [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = -\log_3(x + 3) + 9 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Start by isolating the logarithmic term:
[tex]\[ y - 9 = -\log_3(x + 3) \][/tex]
Multiply both sides by -1:
[tex]\[ 9 - y = \log_3(x + 3) \][/tex]
Rewrite the equation in exponential form to solve for [tex]\( x \)[/tex]:
[tex]\[ 3^{9 - y} = x + 3 \][/tex]
Subtract 3 from both sides:
[tex]\[ x = 3^{9 - y} - 3 \][/tex]
3. State the inverse function:
[tex]\[ f^{-1}(y) = 3^{9 - y} - 3 \][/tex]
4. Find the lattice points:
Lattice points are points [tex]\((y, f^{-1}(y))\)[/tex] where both coordinates are integers. To find such points, substitute integer values for [tex]\( y \)[/tex] into the inverse function and check if the result is also an integer.
Let’s check the values:
- For [tex]\( y = 0 \)[/tex]:
[tex]\[ f^{-1}(0) = 3^{9 - 0} - 3 = 3^9 - 3 = 19683 - 3 = 19680 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 1 \)[/tex]:
[tex]\[ f^{-1}(1) = 3^{9 - 1} - 3 = 3^8 - 3 = 6561 - 3 = 6558 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 2 \)[/tex]:
[tex]\[ f^{-1}(2) = 3^{9 - 2} - 3 = 3^7 - 3 = 2187 - 3 = 2184 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 3 \)[/tex]:
[tex]\[ f^{-1}(3) = 3^{9 - 3} - 3 = 3^6 - 3 = 729 - 3 = 726 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 4 \)[/tex]:
[tex]\[ f^{-1}(4) = 3^{9 - 4} - 3 = 3^5 - 3 = 243 - 3 = 240 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 5 \)[/tex]:
[tex]\[ f^{-1}(5) = 3^{9 - 5} - 3 = 3^4 - 3 = 81 - 3 = 78 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 6 \)[/tex]:
[tex]\[ f^{-1}(6) = 3^{9 - 6} - 3 = 3^3 - 3 = 27 - 3 = 24 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 7 \)[/tex]:
[tex]\[ f^{-1}(7) = 3^{9 - 7} - 3 = 3^2 - 3 = 9 - 3 = 6 \quad \text{(lattice point: } (7, 6) \text{)} \][/tex]
- For [tex]\( y = 8 \)[/tex]:
[tex]\[ f^{-1}(8) = 3^{9 - 8} - 3 = 3^1 - 3 = 3 - 3 = 0 \quad \text{(lattice point: } (8, 0) \text{)} \][/tex]
- For [tex]\( y = 9 \)[/tex]:
[tex]\[ f^{-1}(9) = 3^{9 - 9} - 3 = 3^0 - 3 = 1 - 3 = -2 \quad \text{(lattice point: } (9, -2) \text{)} \][/tex]
So, the lattice points of the inverse function are:
- [tex]\( (7, 6) \)[/tex]
- [tex]\( (8, 0) \)[/tex]
- [tex]\( (9, -2) \)[/tex]
You can plot these points on the graph.
1. Set [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = -\log_3(x + 3) + 9 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Start by isolating the logarithmic term:
[tex]\[ y - 9 = -\log_3(x + 3) \][/tex]
Multiply both sides by -1:
[tex]\[ 9 - y = \log_3(x + 3) \][/tex]
Rewrite the equation in exponential form to solve for [tex]\( x \)[/tex]:
[tex]\[ 3^{9 - y} = x + 3 \][/tex]
Subtract 3 from both sides:
[tex]\[ x = 3^{9 - y} - 3 \][/tex]
3. State the inverse function:
[tex]\[ f^{-1}(y) = 3^{9 - y} - 3 \][/tex]
4. Find the lattice points:
Lattice points are points [tex]\((y, f^{-1}(y))\)[/tex] where both coordinates are integers. To find such points, substitute integer values for [tex]\( y \)[/tex] into the inverse function and check if the result is also an integer.
Let’s check the values:
- For [tex]\( y = 0 \)[/tex]:
[tex]\[ f^{-1}(0) = 3^{9 - 0} - 3 = 3^9 - 3 = 19683 - 3 = 19680 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 1 \)[/tex]:
[tex]\[ f^{-1}(1) = 3^{9 - 1} - 3 = 3^8 - 3 = 6561 - 3 = 6558 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 2 \)[/tex]:
[tex]\[ f^{-1}(2) = 3^{9 - 2} - 3 = 3^7 - 3 = 2187 - 3 = 2184 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 3 \)[/tex]:
[tex]\[ f^{-1}(3) = 3^{9 - 3} - 3 = 3^6 - 3 = 729 - 3 = 726 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 4 \)[/tex]:
[tex]\[ f^{-1}(4) = 3^{9 - 4} - 3 = 3^5 - 3 = 243 - 3 = 240 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 5 \)[/tex]:
[tex]\[ f^{-1}(5) = 3^{9 - 5} - 3 = 3^4 - 3 = 81 - 3 = 78 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 6 \)[/tex]:
[tex]\[ f^{-1}(6) = 3^{9 - 6} - 3 = 3^3 - 3 = 27 - 3 = 24 \quad \text{(not a lattice point)} \][/tex]
- For [tex]\( y = 7 \)[/tex]:
[tex]\[ f^{-1}(7) = 3^{9 - 7} - 3 = 3^2 - 3 = 9 - 3 = 6 \quad \text{(lattice point: } (7, 6) \text{)} \][/tex]
- For [tex]\( y = 8 \)[/tex]:
[tex]\[ f^{-1}(8) = 3^{9 - 8} - 3 = 3^1 - 3 = 3 - 3 = 0 \quad \text{(lattice point: } (8, 0) \text{)} \][/tex]
- For [tex]\( y = 9 \)[/tex]:
[tex]\[ f^{-1}(9) = 3^{9 - 9} - 3 = 3^0 - 3 = 1 - 3 = -2 \quad \text{(lattice point: } (9, -2) \text{)} \][/tex]
So, the lattice points of the inverse function are:
- [tex]\( (7, 6) \)[/tex]
- [tex]\( (8, 0) \)[/tex]
- [tex]\( (9, -2) \)[/tex]
You can plot these points on the graph.
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