https://file.aiquickdraw.com/mathbot/file/1726545462554dwac038q.jpg

Let's work through the questions you have in the image.

### **Test 1 - Matching Terms to Definitions**

From the list of terms provided (Domain, Range, etc.), match the correct term to the given definition.

1. **Set of ordered pairs:** This describes a **relation** (Answer: d).
2. **Set of all first coordinates:** This refers to the **domain** (Answer: a).
3. **Set of all second coordinates:** This refers to the **range** (Answer: b).
4. **Denoted by (f ○ g), defined as (f ○ g)(x) = f(g(x)):** This is the **composite function** (Answer: g).
5. **A relation between a set of inputs having one output each:** This describes a **function** (Answer: e).
6. **The ratio of two polynomial functions where the denominator polynomial is not equal to zero:** This is a **rational function** (Answer: h).
7. **Finding the value of f(x) = ... or y = ... that corresponds to a given value of x:** This refers to **evaluating a function** (Answer: f).
8. **A graphical method of determining whether a curve in the plane represents the graph of a function by visually examining the number of intersections of the curve with vertical lines:** This describes the **vertical line test** (Answer: c).
9. **Given f(x) = 2x + 3, g(x) = 3x + 4, find (f ○ g)(x):** First, substitute \( g(x) \) into \( f(x) \): 
   \[
   f(g(x)) = 2(3x + 4) + 3 = 6x + 8 + 3 = 6x + 11
   \]
10. **Given f(x) = 6x + 4, g(x) = 2x + 7, find (f + g)(x):** This is simply the sum of the two functions:
   \[
   f(x) + g(x) = (6x + 4) + (2x + 7) = 8x + 11
   \]

---

### **Test 2 - Domain and Range**

For each relation given, find the **domain** (all possible x-values) and the **range** (all possible y-values). Here's how to interpret the sets:

1. **\{(1, 0), (4, 9), (7, 8), (9, 10), (6, 5)\}**:
   - Domain: {1, 4, 7, 9, 6}
   - Range: {0, 9, 8, 10, 5}

2. **\{(3, 8), (-2, 8), (-5, 8), (-1, -6), (9, 9)\}**:
   - Domain: {3, -2, -5, -1, 9}
   - Range: {8, -6, 9}

3. **\{(5, -9), (2, -9), (9, 5), (11, -6), (6, 5)\}**:
   - Domain: {5, 2, 9, 11, 6}
   - Range: {-9, 5, -6}

4. **\{(1, 4), (9, 5), (-3, 8), (-11, 12), (9, 16)\}**:
   - Domain: {1, 9, -3, -11}
   - Range: {4, 5, 8, 12, 16}

5. **\{(4, 8), (7, 9), (-9, 7), (-7, -1), (-67, 8), (-56, -46)\}**:
   - Domain: {4, 7, -9, -7, -67, -56}
   - Range: {8, 9, 7, -1, -46}

---

### **Test 3 - Function Evaluation**

1. **Given \( f(x) = 2x^2 + 3x + 5 \), find \( f(4) \):**
   \[
   f(4) = 2(4)^2 + 3(4) + 5 = 2(16) + 12 + 5 = 32 + 12 + 5 = 49
   \]

2. **Given \( f(x) = 3x^2 - 4x + 4 \), find \( f(6) \):**
   \[
   f(6) = 3(6)^2 - 4(6) + 4 = 3(36) - 24 + 4 = 108 - 24 + 4 = 88
   \]

3. **Given \( f(x) = 5x^2 + 18x + 7 \), find \( f(9) \):**
   \[
   f(9) = 5(9)^2 + 18(9) + 7 = 5(81) + 162 + 7 = 405 + 162 + 7 = 574
   \]

4. **Given \( f(x) = 8x^2 - 3x + 5 \), find \( f(10) \):**
   \[
   f(10) = 8(10)^2 - 3(10) + 5 = 8(100) - 30 + 5 = 800 - 30 + 5 = 775
   \]

5. **Given \( f(x) = 9x^2 - 4x - 9 \), find \( f(7) \):**
   \[
   f(7) = 9(7)^2 - 4(7) - 9 = 9(49) - 28 - 9 = 441 - 28 - 9 = 404
   \]

Would you like more details or have any questions about these problems? 

---

### Follow-up Questions:
1. What is the significance of the **domain** in a function, and how is it different from the **range**?
2. How do you evaluate a **composite function** like \( (f \circ g)(x) \)?
3. What is the **vertical line test**, and why is it used to identify functions?
4. Can you explain how to compute \( f(x) + g(x) \) for two given functions?
5. What is a **rational function**, and how does it differ from other types of functions?

### Tip:
Always check whether the domain of a function includes any restrictions, especially in cases involving denominators or square roots.

The image provided contains a general mathematics exam with questions about domains, ranges, function evaluation, and matching mathematical terms to their definitions.

This exam tests knowledge of mathematical functions, including finding domains and ranges, evaluating functions, and matching mathematical concepts like composite functions and rational functions. It is suitable for students in Grades 10-12.

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