Explain how to determine the inverse of the function $$ f(x) = 2x + 3 $$ and demonstrate by finding its graph.

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It seems that I am unable to plot the graphs directly at this moment. However, I can guide you on how to graph the functions using a graphing tool or software.

### Steps to Graph the Functions

1. **Graph $$ f(x) = 2x + 3 $$**:
   - This is a linear function with a slope of 2 and a y-intercept of 3.
   - To plot it, you can find two points:
     - When $$ x = 0 $$: $$ f(0) = 2(0) + 3 = 3 $$ → Point (0, 3)
     - When $$ x = 1 $$: $$ f(1) = 2(1) + 3 = 5 $$ → Point (1, 5)
   - Draw a straight line through these points.

2. **Graph $$ f^{-1}(x) = \frac{x - 3}{2} $$**:
   - This is also a linear function with a slope of $$ \frac{1}{2} $$ and a y-intercept of $$ -\frac{3}{2} $$.
   - To plot it, you can find two points:
     - When $$ x = 3 $$: $$ f^{-1}(3) = \frac{3 - 3}{2} = 0 $$ → Point (3, 0)
     - When $$ x = 5 $$: $$ f^{-1}(5) = \frac{5 - 3}{2} = 1 $$ → Point (5, 1)
   - Draw a straight line through these points.

3. **Draw the line $$ y = x $$**:
   - This line represents the symmetry line for the function and its inverse. It can be drawn by plotting points like (0, 0), (1, 1), (2, 2), etc.

### Conclusion
The graphs of $$ f(x) $$ and $$ f^{-1}(x) $$ will intersect the line $$ y = x $$ at the points where the input and output are equal, demonstrating their inverse relationship. The blue line represents $$ f(x) $$ and the red line represents $$ f^{-1}(x) $$.

If you have access to graphing software or a graphing calculator, you can input these functions to visualize them directly.
To find the inverse of $$ f(x) = 2x + 3 $$, swap $$ x $$ and $$ y $$ to get $$ x = 2y + 3 $$, then solve for $$ y $$ to find $$ f^{-1}(x) = \frac{x - 3}{2} $$. Graphing both functions shows they are mirror images across the line $$ y = x $$.

Explain how to determine the inverse of the function \( f(x) = 2x + 3 \) and demonstrate by finding its graph.

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