The given function is:

$f(x) = \frac{3}{2} |x - 3| + 1$

This is an absolute value function, which involves piecewise behavior depending on whether $x - 3$ is positive or negative. Let's analyze it in two cases:

1. For $x \geq 3$:

In this case, $|x - 3| = x - 3$. Substituting this into the function:

$f(x) = \frac{3}{2} (x - 3) + 1$

Simplifying:

$f(x) = \frac{3}{2}x - \frac{9}{2} + 1$

$f(x) = \frac{3}{2}x - \frac{9}{2} + \frac{2}{2}$

$f(x) = \frac{3}{2}x - \frac{7}{2}$

Thus, for $x \geq 3$, the function is:

$f(x) = \frac{3}{2}x - \frac{7}{2}$

2. For $x < 3$:

In this case, $|x - 3| = -(x - 3) = 3 - x$. Substituting this into the function:

$f(x) = \frac{3}{2} (3 - x) + 1$

Simplifying:

$f(x) = \frac{9}{2} - \frac{3}{2}x + 1$

$f(x) = \frac{9}{2} + \frac{2}{2} - \frac{3}{2}x$

$f(x) = \frac{11}{2} - \frac{3}{2}x$

Thus, for $x < 3$, the function is:

$f(x) = \frac{11}{2} - \frac{3}{2}x$

Final Piecewise Function:

\begin{cases} \frac{3}{2}x - \frac{7}{2} & \text{if } x \geq 3 \\ \frac{11}{2} - \frac{3}{2}x & \text{if } x < 3 \end{cases}$$ Would you like more details, or do you have any questions? ### 5 Related Questions: 1. How do you graph the function $$f(x) = \frac{3}{2}|x - 3| + 1$$? 2. What are the key features of absolute value functions? 3. How do piecewise functions behave at critical points like $$x = 3$$? 4. How does the slope of the lines in the piecewise function change on each interval? 5. What is the range of the function $$f(x) = \frac{3}{2}|x - 3| + 1$$? **Tip:** Absolute value functions typically create "V" shapes on the graph, with the vertex occurring at the point where the expression inside the absolute value equals zero.