To determine the correct statement regarding the function \( f(x) = 4x^4 - 1 \), letβs analyze its properties:
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One-to-One Function: A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once.
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Inverse Function: If a function is not one-to-one, its inverse will not be a function.
Step 1: Analyze the Function
The given function \( f(x) = 4x^4 - 1 \) is a polynomial of degree 4. The form of this polynomial indicates that it has a minimum point (since it is an even-degree polynomial with a positive leading coefficient).
Step 2: Test for One-to-One
To check if it passes the horizontal line test, you can consider the derivative:
\[ f'(x) = 16x^3 \]
The derivative equals zero when \( x = 0 \).
- For \( x < 0 \), \( f'(x) < 0 \) (function is decreasing).
- For \( x > 0 \), \( f'(x) > 0 \) (function is increasing).
Since the function decreases to a minimum at \( x = 0 \) and then increases, it implies the function is not one-to-one (the same value is achieved at different points, e.g., both \( f(-a) \) and \( f(a) \) result in the same value).
Step 3: Conclusion about Inverses
Since \( f(x) \) is not one-to-one, it does not pass the horizontal line test, and therefore, it does not have an inverse that is a function.
Final Answer
The correct response is: The function \( f(x) \) is not one-to-one; thus, the inverse of \( f^{-1}(x) \) is not a function.