$$ f(x)=x^{3}-4 $$ A. The function $$ f(x) $$ is one-to-one and $$ f^{-1} $$ (and equation for the inverse function in B. The function is not one-to-one. (b) Choose the correct graph of $$ f $$ and $$ f^{-1} $$. The

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To determine whether the function $$ f(x) = x^3 - 4 $$ is one-to-one, we can analyze its properties.

### A. Is the function one-to-one?

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.

1. **Monotonicity**: We can check if the function is strictly increasing or strictly decreasing by examining its derivative:
   $$
   f'(x) = 3x^2
   $$
   Since $$ 3x^2 $$ is always non-negative and is equal to zero only at $$ x = 0 $$, the function $$ f(x) $$ is strictly increasing for all $$ x $$. This means that $$ f(x) $$ is one-to-one.

2. **Finding the Inverse**: Since $$ f(x) $$ is one-to-one, we can find its inverse. To find $$ f^{-1}(x) $$, we set $$ y = f(x) $$:
   $$
   y = x^3 - 4
   $$
   Rearranging gives:
   $$
   y + 4 = x^3
   $$
   Taking the cube root of both sides:
   $$
   x = \sqrt[3]{y + 4}
   $$
   Thus, the inverse function is:
   $$
   f^{-1}(x) = \sqrt[3]{x + 4}
   $$

### B. Graph of $$ f $$ and $$ f^{-1} $$

To graph $$ f(x) $$ and $$ f^{-1}(x) $$:

1. **Graph of $$ f(x) = x^3 - 4 $$**:
   - The graph is a cubic function shifted down by 4 units. It passes through the point $$ (0, -4) $$ and has the general shape of a cubic function.

2. **Graph of $$ f^{-1}(x) = \sqrt[3]{x + 4} $$**:
   - The graph of the inverse function is a cube root function shifted left by 4 units. It passes through the point $$ (-4, 0) $$ and has the general shape of a cube root function.

### Summary

- **A**: The function $$ f(x) = x^3 - 4 $$ is one-to-one, and its inverse is $$ f^{-1}(x) = \sqrt[3]{x + 4} $$.
- **B**: The correct graphs would show $$ f(x) $$ as a cubic curve and $$ f^{-1}(x) $$ as a cube root curve, with the two graphs being reflections of each other across the line $$ y = x $$.

If you have specific graph options to choose from, please provide them, and I can help identify the correct one!
The function $$ f(x) = x^3 - 4 $$ is one-to-one, and its inverse is $$ f^{-1}(x) = \sqrt[3]{x + 4} $$. The graphs of $$ f(x) $$ and $$ f^{-1}(x) $$ are reflections of each other across the line $$ y = x $$.

\( f(x)=x^{3}-4 \) A. The function \( f(x) \) is one-to-one and \( f^{-1} \) (and equation for the inverse function in B. The function is not one-to-one. (b) Choose the correct graph of \( f \) and \( f^{-1} \). The

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