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To find the lattice points of the inverse function [tex]\[f^{-1}(x)\][/tex] on a graph, we need to follow a specific process. Here's a step-by-step solution:
1. Understand the Function and Inverse:
- Given the function [tex]\( f(x) = \log_3(-x + 4) + 5 \)[/tex].
- The inverse function essentially swaps the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation. Therefore, we solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] instead of [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
2. Rewrite the Function for the Inverse:
- Start with [tex]\( y = \log_3(-x + 4) + 5 \)[/tex].
- Subtract 5 from both sides to isolate the logarithmic expression:
[tex]\[ y - 5 = \log_3(-x + 4) \][/tex].
- Rewrite the equation in exponential form (since [tex]\(\log_3 b = a\)[/tex] translates to [tex]\(3^a = b\)[/tex]):
[tex]\[ 3^{y - 5} = -x + 4 \][/tex].
3. Solve for [tex]\( x \)[/tex]:
- Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ -x = 3^{y - 5} - 4 \][/tex],
[tex]\[ x = -3^{y - 5} + 4 \][/tex].
4. Identify Lattice Points:
- Now we have the inverse function [tex]\( f^{-1}(x) = -3^{y - 5} + 4 \)[/tex].
- Identify specific values for [tex]\( y \)[/tex] (or [tex]\( x \)[/tex]) that produce integer outputs in the inverse function.
- It's often useful to test small integer values for [tex]\( y \)[/tex] first and compute their corresponding [tex]\( x \)[/tex] values to see if they are integers (lattice points).
5. Calculate and Plot Points:
- For [tex]\( y = 5 \)[/tex]:
[tex]\[ x = -3^{5 - 5} + 4 = -3^0 + 4 = -1 + 4 = 3 \][/tex]
Thus, one lattice point is [tex]\( (3, 5) \)[/tex].
- For [tex]\( y = 6 \)[/tex]:
[tex]\[ x = -3^{6 - 5} + 4 = -3^1 + 4 = -3 + 4 = 1 \][/tex]
Thus, another lattice point is [tex]\( (1, 6) \)[/tex].
6. Continue for Other Values if Needed:
- For [tex]\( y = 4 \)[/tex], checking continuity:
[tex]\[ x = -3^{4 - 5} + 4 = -3^{-1} + 4 \][/tex]
Since [tex]\(3^{-1} = \frac{1}{3}\)[/tex], it’s not an integer.
- Similarly, test other [tex]\( y \)[/tex] values (both positive and negative) to find more lattice points.
Finally, plot the identified lattice points [tex]\((3, 5)\)[/tex] and [tex]\((1, 6)\)[/tex] on the graph, ensuring they are correctly placed. Continue this process to find and plot any other lattice points if they exist.
1. Understand the Function and Inverse:
- Given the function [tex]\( f(x) = \log_3(-x + 4) + 5 \)[/tex].
- The inverse function essentially swaps the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation. Therefore, we solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] instead of [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
2. Rewrite the Function for the Inverse:
- Start with [tex]\( y = \log_3(-x + 4) + 5 \)[/tex].
- Subtract 5 from both sides to isolate the logarithmic expression:
[tex]\[ y - 5 = \log_3(-x + 4) \][/tex].
- Rewrite the equation in exponential form (since [tex]\(\log_3 b = a\)[/tex] translates to [tex]\(3^a = b\)[/tex]):
[tex]\[ 3^{y - 5} = -x + 4 \][/tex].
3. Solve for [tex]\( x \)[/tex]:
- Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ -x = 3^{y - 5} - 4 \][/tex],
[tex]\[ x = -3^{y - 5} + 4 \][/tex].
4. Identify Lattice Points:
- Now we have the inverse function [tex]\( f^{-1}(x) = -3^{y - 5} + 4 \)[/tex].
- Identify specific values for [tex]\( y \)[/tex] (or [tex]\( x \)[/tex]) that produce integer outputs in the inverse function.
- It's often useful to test small integer values for [tex]\( y \)[/tex] first and compute their corresponding [tex]\( x \)[/tex] values to see if they are integers (lattice points).
5. Calculate and Plot Points:
- For [tex]\( y = 5 \)[/tex]:
[tex]\[ x = -3^{5 - 5} + 4 = -3^0 + 4 = -1 + 4 = 3 \][/tex]
Thus, one lattice point is [tex]\( (3, 5) \)[/tex].
- For [tex]\( y = 6 \)[/tex]:
[tex]\[ x = -3^{6 - 5} + 4 = -3^1 + 4 = -3 + 4 = 1 \][/tex]
Thus, another lattice point is [tex]\( (1, 6) \)[/tex].
6. Continue for Other Values if Needed:
- For [tex]\( y = 4 \)[/tex], checking continuity:
[tex]\[ x = -3^{4 - 5} + 4 = -3^{-1} + 4 \][/tex]
Since [tex]\(3^{-1} = \frac{1}{3}\)[/tex], it’s not an integer.
- Similarly, test other [tex]\( y \)[/tex] values (both positive and negative) to find more lattice points.
Finally, plot the identified lattice points [tex]\((3, 5)\)[/tex] and [tex]\((1, 6)\)[/tex] on the graph, ensuring they are correctly placed. Continue this process to find and plot any other lattice points if they exist.
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