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Let's work through each part of the assignment you've uploaded.

### 1. Evaluate \( f(x) \)
The function \( f(x) \) is defined piecewise:
\[
f(x) = 
\begin{cases} 
3x - 4 & \text{if } x \leq 0 \\
x^2 + x + 3 & \text{if } x = 3 \\
\sqrt{x + 5} & \text{if } x \geq 4
\end{cases}
\]
You need to evaluate \( f(x) \) for various values of \( x \):

- **(a) \( x = 6 \)**: Since \( x \geq 4 \), use \( f(x) = \sqrt{x + 5} \).
  \[
  f(6) = \sqrt{6 + 5} = \sqrt{11}
  \]

- **(b) \( x = 3 \)**: Since \( x = 3 \), use \( f(x) = x^2 + x + 3 \).
  \[
  f(3) = 3^2 + 3 + 3 = 9 + 3 + 3 = 15
  \]

- **(c) \( x = -4 \)**: Since \( x \leq 0 \), use \( f(x) = 3x - 4 \).
  \[
  f(-4) = 3(-4) - 4 = -12 - 4 = -16
  \]

- **(d) \( x = 8 \)**: Since \( x \geq 4 \), use \( f(x) = \sqrt{x + 5} \).
  \[
  f(8) = \sqrt{8 + 5} = \sqrt{13}
  \]

- **(e) \( x = -1 \)**: Since \( x \leq 0 \), use \( f(x) = 3x - 4 \).
  \[
  f(-1) = 3(-1) - 4 = -3 - 4 = -7
  \]

### 2. Sketch the graph of \( f(x) = 3x^2 - x + 2 \)
You are asked to sketch the graph of the quadratic function \( f(x) = 3x^2 - x + 2 \). To do this, we need to compute the \( y \)-values for the given \( x \)-values in the table. Let’s fill in the missing values:

- For \( x = -2 \):
  \[
  f(-2) = 3(-2)^2 - (-2) + 2 = 3(4) + 2 + 2 = 12 + 2 + 2 = 16
  \]
  
- For \( x = -1 \):
  \[
  f(-1) = 3(-1)^2 - (-1) + 2 = 3(1) + 1 + 2 = 3 + 1 + 2 = 6
  \]

- For \( x = 0 \):
  \[
  f(0) = 3(0)^2 - 0 + 2 = 2
  \]

- For \( x = 1 \):
  \[
  f(1) = 3(1)^2 - 1 + 2 = 3 - 1 + 2 = 4
  \]

- For \( x = 2 \):
  \[
  f(2) = 3(2)^2 - 2 + 2 = 3(4) - 2 + 2 = 12 - 2 + 2 = 12
  \]

So the table will look like:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 16 \\
-1 & 6 \\
0 & 2 \\
1 & 4 \\
2 & 12 \\
\hline
\end{array}
\]
You can now plot these points to sketch the graph.

### 3. Ordered Pairs and Relation
Looking at the diagram, we can extract the ordered pairs:
\[
(-3, 3), (1, 3), (3, 9)
\]
This represents a **many-to-one** relation, as multiple \( x \)-values map to the same \( y \)-value (i.e., both \( -3 \) and \( 1 \) map to \( 3 \)).

### 4. Input, Function, and Output
From the table:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 7 \\
-1 & 4 \\
0 & 3 \\
1 & 4 \\
2 & 7 \\
\hline
\end{array}
\]
The function that best fits this data would need to be determined, possibly involving quadratic or linear expressions. From a quick inspection, a possible quadratic relation could exist because \( y \) increases then decreases symmetrically.

---

Would you like to dive deeper into any of these parts or need further clarification?

Here are 5 questions to extend your understanding:
1. How do you determine which part of a piecewise function to use for specific values of \( x \)?
2. What strategies can help you quickly identify if a relation is one-to-one or many-to-one?
3. How do you graph quadratic functions like \( 3x^2 - x + 2 \)?
4. Can you derive a function from the given inputs and outputs in problem 4?
5. What does the term "ordered pair" refer to, and why is it important in functions?

**Tip**: Always check the domain of each part of a piecewise function before plugging in values!

1. Evaluate f(x) for different values of x, 2. Sketch the graph of the function f(x) = 3x^2 - x + 2, 3. Write ordered pairs from the diagram and identify the type of relation, 4. Determine the input, function, and output from the table.

Learn how to evaluate a piecewise function f(x) and sketch the graph of a quadratic function f(x) = 3x^2 - x + 2. This covers the evaluation of functions for specific values, graphing quadratic equations, and determining the type of relation from a given diagram. Ideal for Grades 9-11.

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