1. **Problem Statement:** Find the domain and range of the function $f(x) = |x^3 + 1|$. Determine for which values of $x$ the derivative is undefined. Find the inverse function if it exists, otherwise explain why not. 2. **Domain:** The function inside the absolute value is $x^3 + 1$, which is defined for all real $x$. Since absolute value is defined for all real numbers, the domain is: $$\text{Domain} = (-\infty, \infty)$$ 3. **Range:** Since $f(x) = |x^3 + 1|$, the output is always non-negative. The minimum value occurs when $x^3 + 1 = 0 \Rightarrow x = -1$, giving $f(-1) = 0$. As $x \to \pm \infty$, $x^3 + 1 \to \pm \infty$, so $|x^3 + 1| \to \infty$. Thus, $$\text{Range} = [0, \infty)$$ 4. **Derivative and points of non-differentiability:** The derivative of $f(x)$ is given by the chain rule: $$f'(x) = \frac{d}{dx} |x^3 + 1| = \frac{x^3 + 1}{|x^3 + 1|} \cdot 3x^2$$ This derivative is undefined where the inside of the absolute value is zero, i.e., where $x^3 + 1 = 0 \Rightarrow x = -1$. 5. **Inverse function:** To have an inverse, $f$ must be one-to-one (injective). Since $f(x) = |x^3 + 1|$ is not one-to-one (e.g., $f(-2) = |(-2)^3 + 1| = |-8 + 1| = 7$ and $f(0) = |0 + 1| = 1$, but values can repeat due to absolute value), it does not have an inverse function over the entire domain. **Summary:** - Domain: $(-\infty, \infty)$ - Range: $[0, \infty)$ - Derivative undefined at $x = -1$ - No inverse function exists because $f$ is not one-to-one.