\( f(x)=\log _{2}(x+3) \) 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.

Let's solve each part step by step. --- ### **(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:** \[ (13, 4) \] --- ### **(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:** \[ (5, 4) \] --- ### **(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 \( (1, 2) \). --- **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 \((1, \ 2)\).