To analyze the functions \( f(x) = 8x \) and \( g(x) = \log_8(x) \), we first need to understand their characteristics and quickly determine points on the graph of \( g(x) \).
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For \( f(x) = 8x \):
- When \( x = 1 \): \[ f(1) = 8(1) = 8 \] So the point \( (1, 8) \) is correct.
- When \( x = -1 \): \[ f(-1) = 8(-1) = -8 \] Thus, the point \( (-1, -8) \) is actually on the graph, but the earlier provided point \( (-1, 0.125) \) appears to refer to values from \( g(x) \) instead.
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For \( g(x) = \log_8(x) \):
- To find points on \( g(x) \), we can convert the values for \( f(x) \) based on the logarithmic relationship:
- Specifically, if \( f(x) = 8x \) yields \( y \), then we want to find \( x \) such that \( g(y) = \log_8(y) \).
- To find points on \( g(x) \), we can convert the values for \( f(x) \) based on the logarithmic relationship:
Now let's see the given options:
- \( (1, 8) \) and \( (-1, 0.125) \) - \( (-1, 0.125) \) does not match \( g(x) \)'s values.
- \( (-1, -8) \) and \( (1, -0.125) \) - No valid since \( y \) can't be negative for \( g(x) \).
- \( (1, 0.125) \) and \( (-1, 8) \) - Here, \( (1, 0.125) \) makes sense since \( g(0.125) = \log_8(0.125) \), while the second part is from \( f(x) \).
- \( (8, 1) \) and \( (0.125, -1) \) - The first point is valid since \( g(8) = 1 \), but \( (0.125, -1) \) is wrong in terms of \( g(x)\) since it cannot return negative values.
Based on the function \( g(x) \):
- Calculate:
- \( g(8) = \log_8(8) = 1 \)
- \( g(0.125) = \log_8(0.125) = -1 \)
Thus, the better selection based on these evaluations: The correct answer would be:
(8, 1) and (0.125, -1)
This appears valid due to the nature of the logarithmic function mapping valid domains; however, care is needed since:
The final response aligns best with the fact that \( g(8) = 1 \) is correct, despite any misalignment in response structure.