https://file.aiquickdraw.com/mathbot/file/1726184575412usav0mw5.jpg

Let's solve the problems shown in the image. The given piecewise function is:

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

We need to evaluate the function for the given values of \( x \).

1. **Evaluate the function for \( x = 5 \):**

Since \( x = 5 \geq -1 \), we use the second part of the function:

\[ f(5) = 5^3 - 3 = 125 - 3 = 122 \]

2. **Evaluate the function for \( x = -1 \):**

Since \( x = -1 \geq -1 \), we use the second part of the function:

\[ f(-1) = (-1)^3 - 3 = -1 - 3 = -4 \]

3. **Evaluate the function for \( x = 0 \):**

Since \( x = 0 \geq -1 \), we use the second part of the function:

\[ f(0) = 0^3 - 3 = 0 - 3 = -3 \]

4. **Evaluate the function for \( x = -2 \):**

Since \( x = -2 < -1 \), we use the first part of the function:

\[ f(-2) = 7(-2) + 1 = -14 + 1 = -13 \]

5. **Evaluate the function for \( x = 1 \):**

Since \( x = 1 \geq -1 \), we use the second part of the function:

\[ f(1) = 1^3 - 3 = 1 - 3 = -2 \]

Here's a summary of the evaluations:
- \( f(5) = 122 \)
- \( f(-1) = -4 \)
- \( f(0) = -3 \)
- \( f(-2) = -13 \)
- \( f(1) = -2 \)

Would you like details or have any questions about these evaluations? Here are five related questions for further practice:

1. Evaluate the function for \( x = -3 \).
2. Evaluate the function for \( x = 2 \).
3. What is the value of \( f(x) \) when \( x = -0.5 \)?
4. For what values of \( x \) is \( f(x) = 0 \)?
5. Determine the intervals where \( f(x) \) is increasing.

**Tip:** When dealing with piecewise functions, always check the condition for the value of \( x \) to determine which part of the function to use.

Evaluate the function f(x) = {7x + 1, x < -1; x^3 - 3, x >= -1} for x = 5, x = -1, x = 0, x = -2, and x = 1.

Learn how to evaluate a piecewise function with the given expressions for different values of x. This solution covers algebraic concepts, specifically piecewise and polynomial functions. Suitable for students in Grades 9-12.

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