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Question 1209618: Suppose f(x) = \frac{9}{5} x - 4 + x^2. Does f have an inverse? If so, find f^{-1}(0). If not, enter "undef".
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine if f(x) has an inverse and find f⁻¹(0) if it does:
1. **Check if f(x) is one-to-one:**
A function has an inverse if and only if it is one-to-one (injective), meaning that every element in the range corresponds to exactly one element in the domain. A simple way to check if a function is one-to-one is to see if it passes the horizontal line test. If any horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse.
Our function is f(x) = x² + (9/5)x - 4. This is a quadratic function, and its graph is a parabola. Parabolas fail the horizontal line test (except for very special cases where the domain is restricted to only one side of the vertex). Since this is a parabola, it is not one-to-one.
2. **Conclusion:**
Since f(x) is not one-to-one, it does not have an inverse.
Therefore, f⁻¹(0) is *undef*.
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