$$ 1 \leftarrow \quad $$ For $$ f(x)=9 x-6 $$ and $$ g(x)=\frac{x+6}{9} $$, find the following functions. a. $$ (f \circ g)(x) ; $$ b. $$ (g \circ f)(x) ; $$ c. $$ (f \circ g)(5) ; $$ d. $$ (g \circ f)(5) $$ a. $$ (f \circ g)(x)=x $$ (Simplify your answer.) b. $$ (g \circ f)(x)=\square $$ (Simplify your answer.)

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**Problem:**
For $$ f(x) = 9x - 6 $$ and $$ g(x) = \frac{x + 6}{9} $$, find the following functions.

a. $$ (f \circ g)(x) $$  
b. $$ (g \circ f)(x) $$  
c. $$ (f \circ g)(5) $$  
d. $$ (g \circ f)(5) $$

---

### **a. $$ (f \circ g)(x) $$**  
Given:
$$
f(x) = 9x - 6 \quad \text{and} \quad g(x) = \frac{x + 6}{9}
$$

$$
(f \circ g)(x) = f(g(x)) = f\left(\frac{x + 6}{9}\right) = 9\left(\frac{x + 6}{9}\right) - 6 = (x + 6) - 6 = x
$$

**Answer:**  
$$
(f \circ g)(x) = x
$$

---

### **b. $$ (g \circ f)(x) $$**

**Solution:**

$$
(g \circ f)(x) = g(f(x)) = g(9x - 6) = \frac{(9x - 6) + 6}{9} = \frac{9x}{9} = x
$$

**Answer:**  
$$
(g \circ f)(x) = x
$$

---

### **c. $$ (f \circ g)(5) $$**

**Solution:**

Since $$ (f \circ g)(x) = x $$,

$$
(f \circ g)(5) = 5
$$

**Answer:**  
$$
(f \circ g)(5) = 5
$$

---

### **d. $$ (g \circ f)(5) $$**

**Solution:**

Since $$ (g \circ f)(x) = x $$,

$$
(g \circ f)(5) = 5
$$

**Answer:**  
$$
(g \circ f)(5) = 5
$$

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**Summary:**

- a. $$ (f \circ g)(x) = x $$
- b. $$ (g \circ f)(x) = x $$
- c. $$ (f \circ g)(5) = 5 $$
- d. $$ (g \circ f)(5) = 5 $$

Both compositions $$ (f \circ g)(x) $$ and $$ (g \circ f)(x) $$ simplify to the identity function $$ x $$, indicating that $$ f $$ and $$ g $$ are inverse functions of each other.
a. $$ (f \circ g)(x) = x $$ b. $$ (g \circ f)(x) = x $$

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