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Sagot :
To solve the given problem, we need to reflect the function \( f(x) = \frac{1}{6}\left(\frac{2}{5}\right)^x \) across the y-axis. This means we need to find the function \( g(x) = f(-x) \).
First, let's express \( g(x) \):
[tex]\[ g(x) = f(-x) \][/tex]
[tex]\[ g(x) = \frac{1}{6}\left(\frac{2}{5}\right)^{-x} \][/tex]
We can use the property of exponents \( a^{-b} = \frac{1}{a^b} \):
[tex]\[ g(x) = \frac{1}{6} \left(\frac{5}{2}\right)^x \][/tex]
Now, we need to check which of the given ordered pairs lies on the function \( g(x) \):
1. For \((-3, \frac{4}{375})\):
[tex]\[ g(-3) = \frac{1}{6} \left(\frac{5}{2}\right)^{-3} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{5}{2}\right)^{-3} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{2}{5}\right)^3 \][/tex]
[tex]\[ = \frac{1}{6} \cdot \frac{8}{125} \][/tex]
[tex]\[ = \frac{8}{750} \][/tex]
[tex]\[ = \frac{4}{375} \][/tex]
This result matches the given y-value, so \((-3, \frac{4}{375})\) is a valid ordered pair on \( g(x) \).
2. For \((-2, \frac{25}{24})\):
[tex]\[ g(-2) = \frac{1}{6} \left(\frac{5}{2}\right)^{-2} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{2}{5}\right)^2 \][/tex]
[tex]\[ = \frac{1}{6} \cdot \frac{4}{25} \][/tex]
[tex]\[ = \frac{4}{150} \][/tex]
[tex]\[ = \frac{2}{75} \][/tex]
This does not match the given y-value of \(\frac{25}{24}\).
3. For \( (2, \frac{2}{75}) \):
[tex]\[ g(2) = \frac{1}{6} \left(\frac{5}{2}\right)^{2} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{25}{4}\right) \][/tex]
[tex]\[ = \frac{1}{6} \cdot \frac{25}{4} \][/tex]
[tex]\[ = \frac{25}{24} \][/tex]
This does not match the given y-value of \(\frac{2}{75}\).
4. For \( (3, -\frac{125}{48}) \):
[tex]\[ g(3) = \frac{1}{6} \left(\frac{5}{2}\right)^{3} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{125}{8}\right) \][/tex]
[tex]\[ = \frac{125}{48} \][/tex]
This does not match the given y-value of \(-\frac{125}{48}\).
Therefore, the correct ordered pair that lies on \( g(x) \) is:
[tex]\(\boxed{(-3, \frac{4}{375})}\)[/tex].
First, let's express \( g(x) \):
[tex]\[ g(x) = f(-x) \][/tex]
[tex]\[ g(x) = \frac{1}{6}\left(\frac{2}{5}\right)^{-x} \][/tex]
We can use the property of exponents \( a^{-b} = \frac{1}{a^b} \):
[tex]\[ g(x) = \frac{1}{6} \left(\frac{5}{2}\right)^x \][/tex]
Now, we need to check which of the given ordered pairs lies on the function \( g(x) \):
1. For \((-3, \frac{4}{375})\):
[tex]\[ g(-3) = \frac{1}{6} \left(\frac{5}{2}\right)^{-3} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{5}{2}\right)^{-3} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{2}{5}\right)^3 \][/tex]
[tex]\[ = \frac{1}{6} \cdot \frac{8}{125} \][/tex]
[tex]\[ = \frac{8}{750} \][/tex]
[tex]\[ = \frac{4}{375} \][/tex]
This result matches the given y-value, so \((-3, \frac{4}{375})\) is a valid ordered pair on \( g(x) \).
2. For \((-2, \frac{25}{24})\):
[tex]\[ g(-2) = \frac{1}{6} \left(\frac{5}{2}\right)^{-2} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{2}{5}\right)^2 \][/tex]
[tex]\[ = \frac{1}{6} \cdot \frac{4}{25} \][/tex]
[tex]\[ = \frac{4}{150} \][/tex]
[tex]\[ = \frac{2}{75} \][/tex]
This does not match the given y-value of \(\frac{25}{24}\).
3. For \( (2, \frac{2}{75}) \):
[tex]\[ g(2) = \frac{1}{6} \left(\frac{5}{2}\right)^{2} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{25}{4}\right) \][/tex]
[tex]\[ = \frac{1}{6} \cdot \frac{25}{4} \][/tex]
[tex]\[ = \frac{25}{24} \][/tex]
This does not match the given y-value of \(\frac{2}{75}\).
4. For \( (3, -\frac{125}{48}) \):
[tex]\[ g(3) = \frac{1}{6} \left(\frac{5}{2}\right)^{3} \][/tex]
[tex]\[ = \frac{1}{6} \cdot \left(\frac{125}{8}\right) \][/tex]
[tex]\[ = \frac{125}{48} \][/tex]
This does not match the given y-value of \(-\frac{125}{48}\).
Therefore, the correct ordered pair that lies on \( g(x) \) is:
[tex]\(\boxed{(-3, \frac{4}{375})}\)[/tex].
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