To determine whether a function has an inverse, we need to verify whether the function is one-to-one (injective). A function is one-to-one if every output corresponds to exactly one input. For $f(x) = x^3 + x^2$, the issue lies in the lack of one-to-one correspondence, as follows:

1. Failing the Horizontal Line Test

Graphically, a function has an inverse if any horizontal line intersects its graph at most once. The graph of $f(x) = x^3 + x^2$ does not pass this test. For example, horizontal lines will intersect the curve at multiple points because the function is not monotonic (it does not consistently increase or decrease).

2. Derivative Analysis

The derivative $f'(x) = 3x^2 + 2x$ can reveal where the function is increasing or decreasing: $f'(x) = x(3x + 2).$

• The critical points are $x = 0$ and $x = -\frac{2}{3}$.
• The sign of $f'(x)$:
  • For $x < -\frac{2}{3}$, $f'(x) > 0$ (increasing).
  • For $-\frac{2}{3} < x < 0$, $f'(x) < 0$ (decreasing).
  • For $x > 0$, $f'(x) > 0$ (increasing).

This means $f(x)$ increases, decreases, and then increases again, showing it is not monotonic and thus not one-to-one.

3. Example of Non-Injectivity

Consider two distinct inputs:

• $f(0) = 0^3 + 0^2 = 0$,
• $f(-1) = (-1)^3 + (-1)^2 = -1 + 1 = 0$.

Since both $x = 0$ and $x = -1$ map to the same output ($f(0) = f(-1) = 0$), the function is not one-to-one.

Conclusion

The function $f(x) = x^3 + x^2$ does not have an inverse because it fails the criteria of being one-to-one.

Would you like a graph to visualize this behavior, or further clarification?

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Related Questions

1. What does it mean for a function to be monotonic, and why is it necessary for an inverse?
2. How does the horizontal line test work to identify one-to-one functions?
3. Can restricting the domain of $f(x) = x^3 + x^2$ allow it to have an inverse?
4. How do critical points help identify the behavior of a function?
5. What are examples of functions similar to $f(x)$ that do have inverses?

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Tip

When analyzing whether a function has an inverse, always check its derivative to determine monotonicity and use the horizontal line test for a quick graphical check.