https://file.aiquickdraw.com/mathbot/file/1729646001920ubbe4iby.jpg

Let's evaluate the expressions using the table provided.

Given:
- **x** values: \(-3, -2, -1, 0, 1, 2, 3\)
- **f(x)** values: \(10, 8, 6, 4, 2, 0, -2\)
- **g(x)** values: \(-9, -3, 0, 1, 0, -3, -9\)

The questions involve **compositions of functions**, which follow the rule:  
\[(f \circ g)(x) = f(g(x))\quad \text{and}\quad (g \circ f)(x) = g(f(x))\]

Now, let's solve the expressions step-by-step.

---

### a. \((f \circ g)(1)\)
1. Find \(g(1)\): From the table, \(g(1) = 0\).
2. Now, find \(f(0)\): From the table, \(f(0) = 4\).
   - So, \((f \circ g)(1) = f(g(1)) = f(0) = 4\).

---

### b. \((f \circ g)(2)\)
1. Find \(g(2)\): From the table, \(g(2) = -3\).
2. Now, find \(f(-3)\): From the table, \(f(-3) = 10\).
   - So, \((f \circ g)(2) = f(g(2)) = f(-3) = 10\).

---

### c. \((g \circ f)(2)\)
1. Find \(f(2)\): From the table, \(f(2) = 0\).
2. Now, find \(g(0)\): From the table, \(g(0) = 1\).
   - So, \((g \circ f)(2) = g(f(2)) = g(0) = 1\).

---

### d. \((g \circ f)(3)\)
1. Find \(f(3)\): From the table, \(f(3) = -2\).
2. Now, find \(g(-2)\): From the table, \(g(-2) = -3\).
   - So, \((g \circ f)(3) = g(f(3)) = g(-2) = -3\).

---

### e. \((g \circ g)(1)\)
1. Find \(g(1)\): From the table, \(g(1) = 0\).
2. Now, find \(g(0)\): From the table, \(g(0) = 1\).
   - So, \((g \circ g)(1) = g(g(1)) = g(0) = 1\).

---

### f. \((f \circ f)(3)\)
1. Find \(f(3)\): From the table, \(f(3) = -2\).
2. Now, find \(f(-2)\): From the table, \(f(-2) = 8\).
   - So, \((f \circ f)(3) = f(f(3)) = f(-2) = 8\).

---

### Final Answers:
a. \( (f \circ g)(1) = 4 \)  
b. \( (f \circ g)(2) = 10 \)  
c. \( (g \circ f)(2) = 1 \)  
d. \( (g \circ f)(3) = -3 \)  
e. \( (g \circ g)(1) = 1 \)  
f. \( (f \circ f)(3) = 8 \)

---

Would you like to explore any of these steps further?

---

**5 Related Questions:**
1. How do you identify the domain and range of a function?
2. What is the difference between \(f(g(x))\) and \(g(f(x))\)?
3. How do you compose more than two functions, like \(h(f(g(x)))\)?
4. What is the significance of function composition in real-world applications?
5. How can we verify if a function is invertible through composition?

**Tip:**  
When working with function compositions, always solve the inner function first before moving to the outer function.

Evaluate each expression using the values given in the table.

Learn how to evaluate compositions of functions, such as (f ∘ g)(x) and (g ∘ f)(x), using a table of values for f(x) and g(x). This problem explores key algebraic concepts like function composition and is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation