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Sagot :
To determine which table represents the graph of a logarithmic function with both an [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercept., let's translate the provided information into a mathematical context:
A logarithmic function of the form [tex]\( y = \log_b(x) \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex] and typically intersects the [tex]\( y \)[/tex]-axis when [tex]\( x = 1 \)[/tex] because [tex]\( \log_b(1) = 0 \)[/tex]. It generally intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = b^{-y} \)[/tex].
In the given table, we have the following data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0.5 & -0.631 \\ \hline 1.5 & 0.369 \\ \hline 2.5 & 0.834 \\ \hline 3.5 & 1.146 \\ \hline 4.5 & 1.369 \\ \hline \end{array} \][/tex]
From this table, we observe:
- For [tex]\( x = 0.5 \)[/tex], [tex]\( y = -0.631 \)[/tex]
- For [tex]\( x = 1.5 \)[/tex], [tex]\( y = 0.369 \)[/tex]
- For [tex]\( x = 2.5 \)[/tex], [tex]\( y = 0.834 \)[/tex]
- For [tex]\( x = 3.5 \)[/tex], [tex]\( y = 1.146 \)[/tex]
- For [tex]\( x = 4.5 \)[/tex], [tex]\( y = 1.369 \)[/tex]
Let's analyze these pairs:
1. The point [tex]\( (0.5, -0.631) \)[/tex] implies a negative [tex]\( y \)[/tex]-value, which is consistent with the logarithmic function, as [tex]\( \log_b(0.5) \)[/tex] is negative for any base [tex]\( b > 1 \)[/tex].
2. The point [tex]\( (1.5, 0.369) \)[/tex] suggests a positive [tex]\( y \)[/tex]-value, which indicates that for [tex]\( x > 1 \)[/tex], the function value is positive.
3. The value [tex]\( y \)[/tex] is increasing as [tex]\( x \)[/tex] increases, which is also characteristic of a logarithmic function when [tex]\( x > 1 \)[/tex].
Given these points, the table represents a logarithmic function, but we need to confirm if the intercepts coordinate with our bounds.
An intercept understanding:
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 1 \)[/tex], we typically intersect the [tex]\( y \)[/tex]-axis at [tex]\( y = 0 \)[/tex].
- [tex]\( x \)[/tex]-intercept: The table does not explicitly show [tex]\( x > 0 \)[/tex] and [tex]\( x \)[/tex] yielding exactly zero [tex]\( y \)[/tex].
From these, the given intercepts seem consistent with logarithmic characteristics.
Therefore, the table provided does represent the graph of a logarithmic function with both [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts.
A logarithmic function of the form [tex]\( y = \log_b(x) \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex] and typically intersects the [tex]\( y \)[/tex]-axis when [tex]\( x = 1 \)[/tex] because [tex]\( \log_b(1) = 0 \)[/tex]. It generally intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = b^{-y} \)[/tex].
In the given table, we have the following data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0.5 & -0.631 \\ \hline 1.5 & 0.369 \\ \hline 2.5 & 0.834 \\ \hline 3.5 & 1.146 \\ \hline 4.5 & 1.369 \\ \hline \end{array} \][/tex]
From this table, we observe:
- For [tex]\( x = 0.5 \)[/tex], [tex]\( y = -0.631 \)[/tex]
- For [tex]\( x = 1.5 \)[/tex], [tex]\( y = 0.369 \)[/tex]
- For [tex]\( x = 2.5 \)[/tex], [tex]\( y = 0.834 \)[/tex]
- For [tex]\( x = 3.5 \)[/tex], [tex]\( y = 1.146 \)[/tex]
- For [tex]\( x = 4.5 \)[/tex], [tex]\( y = 1.369 \)[/tex]
Let's analyze these pairs:
1. The point [tex]\( (0.5, -0.631) \)[/tex] implies a negative [tex]\( y \)[/tex]-value, which is consistent with the logarithmic function, as [tex]\( \log_b(0.5) \)[/tex] is negative for any base [tex]\( b > 1 \)[/tex].
2. The point [tex]\( (1.5, 0.369) \)[/tex] suggests a positive [tex]\( y \)[/tex]-value, which indicates that for [tex]\( x > 1 \)[/tex], the function value is positive.
3. The value [tex]\( y \)[/tex] is increasing as [tex]\( x \)[/tex] increases, which is also characteristic of a logarithmic function when [tex]\( x > 1 \)[/tex].
Given these points, the table represents a logarithmic function, but we need to confirm if the intercepts coordinate with our bounds.
An intercept understanding:
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 1 \)[/tex], we typically intersect the [tex]\( y \)[/tex]-axis at [tex]\( y = 0 \)[/tex].
- [tex]\( x \)[/tex]-intercept: The table does not explicitly show [tex]\( x > 0 \)[/tex] and [tex]\( x \)[/tex] yielding exactly zero [tex]\( y \)[/tex].
From these, the given intercepts seem consistent with logarithmic characteristics.
Therefore, the table provided does represent the graph of a logarithmic function with both [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts.
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