Direction: Answer the following in a 1 whole sheet of paper. Provide your complete solution.

I. Evaluate the following piecewise functions.

1. f(x) = x + 5 x < - 2; x ^ 2 + 2x + 3 x >= - 2

f(3) =

f(- 4) =

f(- 2) =

Let's evaluate the given piecewise function step by step. The function \( f(x) \) is defined as:

\[
f(x) = \begin{cases}
x + 5 & \text{if } x < -2 \\
x^2 + 2x + 3 & \text{if } x \geq -2
\end{cases}
\]

Now, let's find the values for \( f(3) \), \( f(-4) \), and \( f(-2) \).

### 1. \( f(3) \)
Since \( 3 \geq -2 \), we use the second piece of the function:

\[
f(x) = x^2 + 2x + 3
\]

Substitute \( x = 3 \) into the formula:

\[
f(3) = 3^2 + 2(3) + 3 = 9 + 6 + 3 = 18
\]

Thus, \( f(3) = 18 \).

### 2. \( f(-4) \)
Since \( -4 < -2 \), we use the first piece of the function:

\[
f(x) = x + 5
\]

Substitute \( x = -4 \) into the formula:

\[
f(-4) = -4 + 5 = 1
\]

Thus, \( f(-4) = 1 \).

### 3. \( f(-2) \)
Since \( -2 \geq -2 \), we use the second piece of the function:

\[
f(x) = x^2 + 2x + 3
\]

Substitute \( x = -2 \) into the formula:

\[
f(-2) = (-2)^2 + 2(-2) + 3 = 4 - 4 + 3 = 3
\]

Thus, \( f(-2) = 3 \).

### Final Answers:
- \( f(3) = 18 \)
- \( f(-4) = 1 \)
- \( f(-2) = 3 \)

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Would you like more details or have any questions? Here are 5 related problems for practice:

1. Evaluate \( f(0) \) using the same piecewise function.
2. Evaluate \( f(-3) \) using the same piecewise function.
3. Find \( f(x) \) for \( x = 1 \).
4. What is the value of \( f(x) \) when \( x = -5 \)?
5. For what values of \( x \) does \( f(x) = 3 \)?

**Tip**: When working with piecewise functions, always check which part of the function applies based on the given value of \( x \).

Learn how to evaluate piecewise functions with step-by-step solutions. This problem covers evaluating a piecewise function including linear and quadratic expressions. Suitable for Grades 9-12 students learning about piecewise functions.

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