Step 1: Verifying if $F(x) = x^3 + 3x^2 + 3x + 1$ is One-to-One

To show that $F(x)$ is a one-to-one function, we need to verify that $F'(x) > 0$ for all $x$ in its domain $[-1, \infty)$. This would mean that $F(x)$ is strictly increasing, which implies that $F(x)$ is one-to-one.

1. Find the derivative of $F(x)$:

$F(x) = x^3 + 3x^2 + 3x + 1$ $F'(x) = 3x^2 + 6x + 3$ Factoring out the common factor of 3: $F'(x) = 3(x^2 + 2x + 1)$ $F'(x) = 3(x + 1)^2$

Since $(x + 1)^2 \geq 0$ for all $x$, and it only equals 0 when $x = -1$, $F'(x) \geq 0$ for $x \geq -1$. In fact, $F'(x) > 0$ for all $x > -1$.

Therefore, $F(x)$ is strictly increasing on its domain $[-1, \infty)$. This confirms that $F(x)$ is one-to-one.

Step 2: Finding the Inverse Function $F^{-1}(x)$

Let $y = F(x)$: $y = x^3 + 3x^2 + 3x + 1$ Our goal is to express $x$ in terms of $y$.

Given that this is a cubic equation, finding the exact inverse analytically can be challenging. However, we can guess the form based on simple substitutions, given that:

If $x = -1$, then $F(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0$ Hence, $F(-1) = 0$. The function at $x = -1$ simplifies nicely.

For another common substitution: If $x = 0$, $F(0) = 1$ From this and the fact that the inverse function must pass through these known points, numerical or graphical methods may be employed to find an explicit inverse function expression. However, without a straightforward analytical simplification for arbitrary $x$, the exact algebraic form can be cumbersome.

Nevertheless, the general structure of $F^{-1}(y)$ can be stated as satisfying: $y = x^3 + 3x^2 + 3x + 1$ where $y \in [0, \infty)$ and $x \in [-1, \infty)$.

Step 3: Domain and Range of $F^{-1}$

Since $F(x)$ maps from $[-1, \infty) \rightarrow [0, \infty)$, the inverse function $F^{-1}(x)$ has the domain $[0, \infty)$ and the range $[-1, \infty)$.

Summary:

• Domain of $F(x)$: $[-1, \infty)$
• Range of $F(x)$: $[0, \infty)$
• Domain of $F^{-1}(x)$: $[0, \infty)$
• Range of $F^{-1}(x)$: $[-1, \infty)$

Would you like a more detailed explanation or help with a specific step?

Related Questions:

1. How do you determine if a polynomial is one-to-one using the derivative?
2. Why is the derivative test important for checking if a function is strictly increasing or decreasing?
3. How do you generally find the inverse of a polynomial function?
4. What are the steps to graphically confirm the inverse of a function?
5. Can every function have an inverse, and under what conditions?

Tip: The derivative of a function tells you about the function's slope, which is crucial for determining whether a function is one-to-one.