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Sagot :
To find the inverse of the function \( f(x) = \frac{1}{x^6} \) with \( x > 0 \), we can follow these steps:
### Finding the Inverse Function \( f^{-1}(x) \)
1. Express \( y \) in terms of \( x \):
Given \( y = f(x) = \frac{1}{x^6} \).
2. Swap \( x \) and \( y \):
To find the inverse, we interchange \( x \) and \( y \) to obtain:
[tex]\[ x = \frac{1}{y^6} \][/tex]
3. Solve for \( y \):
Isolate \( y \) by solving:
[tex]\[ y^6 = \frac{1}{x} \][/tex]
Take the sixth root of both sides:
[tex]\[ y = \left( \frac{1}{x} \right)^{\frac{1}{6}} = x^{-\frac{1}{6}} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = x^{-\frac{1}{6}} \][/tex]
### Domain and Range of \( f^{-1}(x) \)
For the inverse function \( f^{-1}(x) = x^{-\frac{1}{6}} \):
- Domain: The domain of \( f^{-1}(x) \) corresponds to the range of the original function \( f(x) \). Since \( f(x) = \frac{1}{x^6} \) for \( x > 0 \), \( f(x) \) is always positive. Therefore, the domain of \( f^{-1}(x) \) is \( x > 0 \).
- Range: The range of \( f^{-1}(x) \) corresponds to the domain of the original function \( f(x) \). Since the domain of \( f(x) \) is \( x > 0 \), the range of \( f^{-1}(x) \) is also \( y > 0 \).
### Verification of the Inverse Function
To verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \), we must show that:
1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)
#### 1. Verifying \( f(f^{-1}(x)) = x \):
[tex]\[ f(f^{-1}(x)) = f(x^{-\frac{1}{6}}) = \frac{1}{(x^{-\frac{1}{6}})^6} \][/tex]
Since:
[tex]\[ (x^{-\frac{1}{6}})^6 = x^{-1} *6 = x^{-6/6} = x^{-1} = x^{1} \][/tex]
Thus:
[tex]\[ f(f^{-1}(x)) = \frac{1}{x^{1}} = x \][/tex]
#### 2. Verifying \( f^{-1}(f(x)) = x \):
[tex]\[ f^{-1}(f(x)) = f^{-1}\left( \frac{1}{x^6} \right) = \left( \frac{1}{x^6} \right)^{-\frac{1}{6}} \][/tex]
Which simplifies to:
[tex]\[ \left( \frac{1}{x^6} \right)^{-\frac{1}{6}} = (x^{-6})^{-\frac{1}{6}} = x \][/tex]
Both conditions \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) are satisfied, confirming that the inverse function is correct.
### Summary
- The inverse function is \( f^{-1}(x) = x^{-\frac{1}{6}} \).
- The domain of \( f^{-1}(x) \) is \( x > 0 \).
- The range of \( f^{-1}(x) \) is \( y > 0 \).
Thus, the solution to the problem is:
[tex]\[ f^{-1}(x) = x^{-\frac{1}{6}}, \quad \text{with domain:} \, x > 0, \quad \text{and range:} \, y > 0. \][/tex]
### Finding the Inverse Function \( f^{-1}(x) \)
1. Express \( y \) in terms of \( x \):
Given \( y = f(x) = \frac{1}{x^6} \).
2. Swap \( x \) and \( y \):
To find the inverse, we interchange \( x \) and \( y \) to obtain:
[tex]\[ x = \frac{1}{y^6} \][/tex]
3. Solve for \( y \):
Isolate \( y \) by solving:
[tex]\[ y^6 = \frac{1}{x} \][/tex]
Take the sixth root of both sides:
[tex]\[ y = \left( \frac{1}{x} \right)^{\frac{1}{6}} = x^{-\frac{1}{6}} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = x^{-\frac{1}{6}} \][/tex]
### Domain and Range of \( f^{-1}(x) \)
For the inverse function \( f^{-1}(x) = x^{-\frac{1}{6}} \):
- Domain: The domain of \( f^{-1}(x) \) corresponds to the range of the original function \( f(x) \). Since \( f(x) = \frac{1}{x^6} \) for \( x > 0 \), \( f(x) \) is always positive. Therefore, the domain of \( f^{-1}(x) \) is \( x > 0 \).
- Range: The range of \( f^{-1}(x) \) corresponds to the domain of the original function \( f(x) \). Since the domain of \( f(x) \) is \( x > 0 \), the range of \( f^{-1}(x) \) is also \( y > 0 \).
### Verification of the Inverse Function
To verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \), we must show that:
1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)
#### 1. Verifying \( f(f^{-1}(x)) = x \):
[tex]\[ f(f^{-1}(x)) = f(x^{-\frac{1}{6}}) = \frac{1}{(x^{-\frac{1}{6}})^6} \][/tex]
Since:
[tex]\[ (x^{-\frac{1}{6}})^6 = x^{-1} *6 = x^{-6/6} = x^{-1} = x^{1} \][/tex]
Thus:
[tex]\[ f(f^{-1}(x)) = \frac{1}{x^{1}} = x \][/tex]
#### 2. Verifying \( f^{-1}(f(x)) = x \):
[tex]\[ f^{-1}(f(x)) = f^{-1}\left( \frac{1}{x^6} \right) = \left( \frac{1}{x^6} \right)^{-\frac{1}{6}} \][/tex]
Which simplifies to:
[tex]\[ \left( \frac{1}{x^6} \right)^{-\frac{1}{6}} = (x^{-6})^{-\frac{1}{6}} = x \][/tex]
Both conditions \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) are satisfied, confirming that the inverse function is correct.
### Summary
- The inverse function is \( f^{-1}(x) = x^{-\frac{1}{6}} \).
- The domain of \( f^{-1}(x) \) is \( x > 0 \).
- The range of \( f^{-1}(x) \) is \( y > 0 \).
Thus, the solution to the problem is:
[tex]\[ f^{-1}(x) = x^{-\frac{1}{6}}, \quad \text{with domain:} \, x > 0, \quad \text{and range:} \, y > 0. \][/tex]
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