Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.