https://file.aiquickdraw.com/mathbot/file/1727314554311bdlhwg7r.jpg

Let's go through each question one by one from the image and solve them accordingly:

---

### **1. Given functions \( f(x) = 2x + 2 \) and \( g(x) = x^2 - 3 \):**

#### a) Find \( (f \circ g)(x) \) (composition of \( f \) and \( g \)):

- \( (f \circ g)(x) \) means \( f(g(x)) \).
  
  \[
  f(g(x)) = f(x^2 - 3) = 2(x^2 - 3) + 2 = 2x^2 - 6 + 2 = 2x^2 - 4
  \]

  So, \( (f \circ g)(x) = 2x^2 - 4 \).

#### b) Find \( (g \circ f)(x) \) (composition of \( g \) and \( f \)):

- \( (g \circ f)(x) \) means \( g(f(x)) \).
  
  \[
  g(f(x)) = g(2x + 2) = (2x + 2)^2 - 3 = (4x^2 + 8x + 4) - 3 = 4x^2 + 8x + 1
  \]

  So, \( (g \circ f)(x) = 4x^2 + 8x + 1 \).

---

### **2. Explain why \( (f \circ g)(x) \) is not always the same as \( (g \circ f)(x) \), and give an example:**

- The composition of functions depends on the order of operations. When composing \( f \) and \( g \), the operations performed on \( x \) are different. For example, in \( (f \circ g)(x) \), the function \( g(x) \) is applied first, followed by \( f \), while in \( (g \circ f)(x) \), \( f(x) \) is applied first.

**Example:**
From question 1, we have:
- \( (f \circ g)(x) = 2x^2 - 4 \)
- \( (g \circ f)(x) = 4x^2 + 8x + 1 \)

Clearly, \( 2x^2 - 4 \neq 4x^2 + 8x + 1 \), so the two compositions are not the same.

---

### **3. Given \( f(x) = 2x + 3 \) and \( g(x) = x - 1 \), find \( (f \circ g)^{-1}(x) \) (the inverse of the composition of \( f \) and \( g \)):**

1. First, find \( (f \circ g)(x) \):

   \[
   (f \circ g)(x) = f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x - 2 + 3 = 2x + 1
   \]

2. Now, find the inverse of \( f \circ g \):

   Set \( y = 2x + 1 \), and solve for \( x \):
   \[
   y = 2x + 1
   \]
   \[
   y - 1 = 2x
   \]
   \[
   x = \frac{y - 1}{2}
   \]

Thus, the inverse is:
\[
(f \circ g)^{-1}(x) = \frac{x - 1}{2}
\]

---

### **4. If \( f(x) = 3x - 2 \) and \( (f \circ g)(x) = 6x + 1 \), find \( g(x) \):**

1. Start by expressing \( (f \circ g)(x) \):
   \[
   (f \circ g)(x) = f(g(x)) = 3g(x) - 2
   \]
   Given that \( (f \circ g)(x) = 6x + 1 \), we have:
   \[
   3g(x) - 2 = 6x + 1
   \]

2. Solve for \( g(x) \):

   \[
   3g(x) = 6x + 1 + 2 = 6x + 3
   \]
   \[
   g(x) = \frac{6x + 3}{3} = 2x + 1
   \]

So, \( g(x) = 2x + 1 \).

---

### **5. From problem 1, find:**

#### a) \( (f \circ g)(2) \):

Recall from problem 1 that \( (f \circ g)(x) = 2x^2 - 4 \).
Now, substitute \( x = 2 \):
\[
(f \circ g)(2) = 2(2^2) - 4 = 2(4) - 4 = 8 - 4 = 4
\]

#### b) \( (g \circ f)(-2) \):

Recall from problem 1 that \( (g \circ f)(x) = 4x^2 + 8x + 1 \).
Now, substitute \( x = -2 \):
\[
(g \circ f)(-2) = 4(-2)^2 + 8(-2) + 1 = 4(4) - 16 + 1 = 16 - 16 + 1 = 1
\]

---

Let me know if you need further details or explanations!

---

### **Related Questions:**

1. Can you explain the step-by-step process of finding the inverse of a composition of functions?
2. What are the main conditions for two functions to be inverse to each other?
3. How does the domain of a composite function differ from that of individual functions?
4. What are some real-life applications of function compositions?
5. Can you provide an example where the composition of two functions results in a constant function?

---

### **Tip:**
When finding the composition of two functions, always apply the second function first and then the first function to avoid errors.

Learn how to solve function compositions such as f(g(x)) and g(f(x)), and find inverses of composed functions. This problem set covers algebraic concepts, including function composition, quadratic functions, and inverse functions, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation