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Sagot :
To find and plot the lattice points of the inverse of the function [tex]\( f(x) = -\log_3(-x + 1) - 4 \)[/tex], we must first determine the inverse function [tex]\( f^{-1}(y) \)[/tex].
Let's start with the given function:
[tex]\[ y = -\log_3(-x + 1) - 4 \][/tex]
To find the inverse, solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
1. Add 4 to both sides:
[tex]\[ y + 4 = -\log_3(-x + 1) \][/tex]
2. Multiply both sides by -1:
[tex]\[ -(y + 4) = \log_3(-x + 1) \][/tex]
3. Rewrite in exponential form to remove the logarithm:
[tex]\[ 3^{-(y + 4)} = -x + 1 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 - 3^{-(y + 4)} \][/tex]
The inverse function is:
[tex]\[ f^{-1}(y) = 1 - 3^{-(y + 4)} \][/tex]
Next, we compute the lattice points (points with integer coordinates) from the inverse function.
Let's find the values of [tex]\( x = f^{-1}(y) \)[/tex] for integer [tex]\( y \)[/tex]:
Here are the lattice points calculated for integer values of [tex]\( y \)[/tex]:
1. For [tex]\( y = -10 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-10 + 4)} = 1 - 3^{-(-6)} = 1 - 3^6 = 1 - 729 = -728 \][/tex]
Lattice point: [tex]\((-728, -10)\)[/tex]
2. For [tex]\( y = -9 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-9 + 4)} = 1 - 3^{-(-5)} = 1 - 3^5 = 1 - 243 = -242 \][/tex]
Lattice point: [tex]\((-242, -9)\)[/tex]
3. For [tex]\( y = -8 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-8 + 4)} = 1 - 3^{-(-4)} = 1 - 3^4 = 1 - 81 = -80 \][/tex]
Lattice point: [tex]\((-80, -8)\)[/tex]
4. For [tex]\( y = -7 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-7 + 4)} = 1 - 3^{-(-3)} = 1 - 3^3 = 1 - 27 = -26 \][/tex]
Lattice point: [tex]\((-26, -7)\)[/tex]
5. For [tex]\( y = -6 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-6 + 4)} = 1 - 3^{-(-2)} = 1 - 3^2 = 1 - 9 = -8 \][/tex]
Lattice point: [tex]\((-8, -6)\)[/tex]
6. For [tex]\( y = -5 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-5 + 4)} = 1 - 3^{-(-1)} = 1 - 3^1 = 1 - 3 = -2 \][/tex]
Lattice point: [tex]\((-2, -5)\)[/tex]
7. For [tex]\( y = -4 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-4 + 4)} = 1 - 3^{-0} = 1 - 1 = 0 \][/tex]
Lattice point: [tex]\((0, -4)\)[/tex]
We then plot these lattice points on the graph:
[tex]\[ (-728, -10), (-242, -9), (-80, -8), (-26, -7), (-8, -6), (-2, -5), (0, -4) \][/tex]
Make sure to mark each of these points on the graph as specified!
Let's start with the given function:
[tex]\[ y = -\log_3(-x + 1) - 4 \][/tex]
To find the inverse, solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
1. Add 4 to both sides:
[tex]\[ y + 4 = -\log_3(-x + 1) \][/tex]
2. Multiply both sides by -1:
[tex]\[ -(y + 4) = \log_3(-x + 1) \][/tex]
3. Rewrite in exponential form to remove the logarithm:
[tex]\[ 3^{-(y + 4)} = -x + 1 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 - 3^{-(y + 4)} \][/tex]
The inverse function is:
[tex]\[ f^{-1}(y) = 1 - 3^{-(y + 4)} \][/tex]
Next, we compute the lattice points (points with integer coordinates) from the inverse function.
Let's find the values of [tex]\( x = f^{-1}(y) \)[/tex] for integer [tex]\( y \)[/tex]:
Here are the lattice points calculated for integer values of [tex]\( y \)[/tex]:
1. For [tex]\( y = -10 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-10 + 4)} = 1 - 3^{-(-6)} = 1 - 3^6 = 1 - 729 = -728 \][/tex]
Lattice point: [tex]\((-728, -10)\)[/tex]
2. For [tex]\( y = -9 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-9 + 4)} = 1 - 3^{-(-5)} = 1 - 3^5 = 1 - 243 = -242 \][/tex]
Lattice point: [tex]\((-242, -9)\)[/tex]
3. For [tex]\( y = -8 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-8 + 4)} = 1 - 3^{-(-4)} = 1 - 3^4 = 1 - 81 = -80 \][/tex]
Lattice point: [tex]\((-80, -8)\)[/tex]
4. For [tex]\( y = -7 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-7 + 4)} = 1 - 3^{-(-3)} = 1 - 3^3 = 1 - 27 = -26 \][/tex]
Lattice point: [tex]\((-26, -7)\)[/tex]
5. For [tex]\( y = -6 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-6 + 4)} = 1 - 3^{-(-2)} = 1 - 3^2 = 1 - 9 = -8 \][/tex]
Lattice point: [tex]\((-8, -6)\)[/tex]
6. For [tex]\( y = -5 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-5 + 4)} = 1 - 3^{-(-1)} = 1 - 3^1 = 1 - 3 = -2 \][/tex]
Lattice point: [tex]\((-2, -5)\)[/tex]
7. For [tex]\( y = -4 \)[/tex]:
[tex]\[ x = 1 - 3^{-(-4 + 4)} = 1 - 3^{-0} = 1 - 1 = 0 \][/tex]
Lattice point: [tex]\((0, -4)\)[/tex]
We then plot these lattice points on the graph:
[tex]\[ (-728, -10), (-242, -9), (-80, -8), (-26, -7), (-8, -6), (-2, -5), (0, -4) \][/tex]
Make sure to mark each of these points on the graph as specified!
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