$$ f(x)=\log _{2}(x+3) $$ and $$ g(x)=\log _{2}(3 x+1) $$. (a) Solve $$ f(x)=4 $$. What point is on the graph of $$ t $$ ? (b) Solve $$ g(x)=4 $$. What point is on the graph of $$ g $$ ? (c) Solve $$ f(x)=g(x) $$. Do the graphs of $$ f $$ and $$ g $$ intersect? If so, where? (d) Solve $$ (f+g)(x)=7 $$. (e) Solve $$ (f-g)(x)=5 $$. (a) The solution set for $$ f(x)=4 $$ is \{ $$ \} $$. (Type an integer or a simplified fraction.) The point where $$ f(x)=4 $$ is $$ \square $$. (Type an ordered pair.) (b) The solution set for $$ g(x)=4 $$ is \{ \}. (Type an integer or a simplified fraction.) The point where $$ g(x)=4 $$ is $$ \square $$. (Type an ordered pair.) (c) The solution set for $$ f(x)=g(x) $$ is $$ \{\square\} $$. (Type an integer or a simplified fraction.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Yes, the graphs do intersect at $$ \square $$. (Type an ordered pair.) B. No, the graphs do not intersect each other.

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Let's solve each part step by step.

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### **(a) Solve $$ f(x) = 4 $$ and Identify the Point on the Graph of $$ f $$**

**Given:**
$$ f(x) = \log_{2}(x + 3) $$
**Equation to Solve:**
$$ \log_{2}(x + 3) = 4 $$

**Solution:**
1. Rewrite the logarithmic equation in exponential form:
   $$ x + 3 = 2^4 $$
2. Calculate $$ 2^4 $$:
   $$ x + 3 = 16 $$
3. Solve for $$ x $$:
   $$ x = 16 - 3 $$
   $$ x = 13 $$

**Solution Set:**
$$ \{13\} $$

**Point on the Graph:**
$$ (13, 4) $$

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### **(b) Solve $$ g(x) = 4 $$ and Identify the Point on the Graph of $$ g $$**

**Given:**
$$ g(x) = \log_{2}(3x + 1) $$
**Equation to Solve:**
$$ \log_{2}(3x + 1) = 4 $$

**Solution:**
1. Rewrite the logarithmic equation in exponential form:
   $$ 3x + 1 = 2^4 $$
2. Calculate $$ 2^4 $$:
   $$ 3x + 1 = 16 $$
3. Solve for $$ x $$:
   $$ 3x = 16 - 1 $$
   $$ 3x = 15 $$
   $$ x = 5 $$

**Solution Set:**
$$ \{5\} $$

**Point on the Graph:**
$$ (5, 4) $$

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### **(c) Solve $$ f(x) = g(x) $$ and Determine if the Graphs Intersect**

**Equations:**
$$ f(x) = \log_{2}(x + 3) $$
$$ g(x) = \log_{2}(3x + 1) $$

**Set the Equations Equal:**
$$ \log_{2}(x + 3) = \log_{2}(3x + 1) $$

**Solution:**
1. Since the logarithmic functions are equal, their arguments must be equal:
   $$ x + 3 = 3x + 1 $$
2. Solve for $$ x $$:
   $$ x + 3 = 3x + 1 $$
   $$ 3 - 1 = 3x - x $$
   $$ 2 = 2x $$
   $$ x = 1 $$
3. Find the corresponding $$ y $$-value using $$ f(x) $$ or $$ g(x) $$:
   $$ f(1) = \log_{2}(1 + 3) = \log_{2}(4) = 2 $$
   $$ g(1) = \log_{2}(3(1) + 1) = \log_{2}(4) = 2 $$

**Solution Set:**
$$ \{1\} $$

**Do the Graphs Intersect?**

**Answer:**

**A.** Yes, the graphs do intersect at $$ (1, 2) $$.

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**Summary of Answers:**

- **(a)**  
  - **Solution Set:** $$\{13\}$$  
  - **Point:** $$(13, \ 4)$$

- **(b)**  
  - **Solution Set:** $$\{5\}$$  
  - **Point:** $$(5, \ 4)$$

- **(c)**  
  - **Solution Set:** $$\{1\}$$  
  - **Conclusion:**  
    **A.** Yes, the graphs do intersect at $$(1, \ 2)$$.
**(a)** - **Solution Set:** $$\{13\}$$ - **Point:** $$(13, 4)$$ **(b)** - **Solution Set:** $$\{5\}$$ - **Point:** $$(5, 4)$$ **(c)** - **Solution Set:** $$\{1\}$$ - **Conclusion:** **A.** Yes, the graphs intersect at $$(1, 2)$$.

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