Let `f(x)={{:(x^(3)-1",", x lt2),(x^(2)+3"," , x ge 2):}` Then

Options

A. `f^(-1)(x)={{:((x+1)^(1//3)",", x lt2),((x-3)^(1//2)+"," , x ge 2):}`
B. `f^(-1)(x)={{:((x+1)^(1//3)",", x lt 7),((x-3)^(1//2)+"," , x ge 7):}`
C. `f^(-1)(x)={{:((x+1)^(1//3)",", x lt 1),((x-3)^(1//2)+"," , x ge 7):}`
D. `f^(-1)(x)` does not exist

Answer

To find the inverse of the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} x^3 - 1 & \text{if } x < 2 \\ x^2 + 3 & \text{if } x \geq 2 \end{cases} \]

we will analyze each piece of the function separately.

### Step 1: Define the function and set \( f(x) = y \)

We start by setting \( f(x) = y \):

1. For \( x < 2 \):
\[ y = x^3 - 1 \]

2. For \( x \geq 2 \):
\[ y = x^2 + 3 \]

### Step 2: Solve for \( x \) in terms of \( y \)

**For the first case** \( (x < 2) \):
\[ y + 1 = x^3 \implies x = (y + 1)^{1/3} \]

**For the second case** \( (x \geq 2) \):
\[ y - 3 = x^2 \implies x = \sqrt{y - 3} \]

### Step 3: Determine the ranges of \( y \)

1. **For \( x < 2 \)**:
- The maximum value of \( f(x) \) occurs at \( x = 2 \):
\[ f(2) = 2^3 - 1 = 8 - 1 = 7 \]
- Therefore, for \( x < 2 \), \( y < 7 \).

2. **For \( x \geq 2 \)**:
- The minimum value of \( f(x) \) occurs at \( x = 2 \):
\[ f(2) = 2^2 + 3 = 4 + 3 = 7 \]
- Therefore, for \( x \geq 2 \), \( y \geq 7 \).

### Step 4: Write the inverse function

Now we can write the inverse function \( f^{-1}(x) \):

\[ f^{-1}(x) = \begin{cases} (x + 1)^{1/3} & \text{if } x < 7 \\ \sqrt{x - 3} & \text{if } x \geq 7 \end{cases} \]

### Conclusion

Thus, the inverse function \( f^{-1}(x) \) is defined as:

\[ f^{-1}(x) = \begin{cases} (x + 1)^{1/3} & \text{if } x < 7 \\ \sqrt{x - 3} & \text{if } x \geq 7 \end{cases} \]