Math Problem Statement
Use the Inverse Function Theorem (Theorem 2.3 in the lecture notes) to show that f and g are inverses of each other. f (x) = x^3 + 1 g(x) = cubic root of √(x − 1)
Solution
To show that and are inverses of each other using the Inverse Function Theorem, we'll follow these steps:
1. Check the compositions: and
For two functions to be inverses, their compositions must satisfy:
Let's compute each composition:
We are given . Now, let's substitute into :
[ f(g(x)) = f\left(\sqrt[3]{x - 1}\right) = \left( \sqrt[3]{x - 1} \right)^3 + 1. ]
Using the property of cube roots, . Thus:
Now, substitute into :
[ g(f(x)) = g(x^3 + 1) = \sqrt[3]{(x^3 + 1) - 1} = \sqrt[3]{x^3}. ]
Since , we have:
2. Apply the Inverse Function Theorem
The Inverse Function Theorem states that if is continuously differentiable and its derivative is nonzero at a point, then has a local inverse near that point. Moreover, the derivative of the inverse function at a point can be expressed as:
Check if is nonzero
First, compute , the derivative of :
Since is nonzero for all , is invertible for by the Inverse Function Theorem.
Verify using the Inverse Function Theorem
If and are inverses, we expect .
- First, find . Since , use the chain rule:
- Now, check :
Since , substitute :
Thus:
This matches , confirming that and are indeed inverses of each other by the Inverse Function Theorem.
Conclusion
We have shown that and are inverses of each other both by checking their compositions and verifying the conditions of the Inverse Function Theorem.
Would you like further details or have any specific questions on this?
Here are some related questions:
- How does the Inverse Function Theorem guarantee the existence of an inverse?
- What happens if at a certain point?
- Can we apply the Inverse Function Theorem to functions of multiple variables?
- What role does continuity play in the Inverse Function Theorem?
- How would the analysis change if had a more complex form, like ?
Tip: When using the Inverse Function Theorem, always ensure that the derivative of the function is nonzero at the points of interest.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Derivatives
Compositions of Functions
Formulas
f(x) = x^3 + 1
g(x) = \sqrt[3]{x - 1}
f'(x) = 3x^2
g'(x) = \frac{1}{3} (x - 1)^{-\frac{2}{3}}
g'(x) = \frac{1}{f'(g(x))}
Theorems
Inverse Function Theorem
Suitable Grade Level
Undergraduate (Calculus II or III)
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