We are given a piecewise-defined function in a graph. We need to express the function in function notation and determine the x- and y-intercepts.

1. Problem Description

We are given a piecewise-defined function in a graph. We need to express the function in function notation and determine the x- and y-intercepts.

2. Solution Steps

First, let's find the equation of each line segment. The break point is at $x=1$.

For $x<1$, the line passes through $(-3, 2)$ and $(1, 4)$. The slope of the line is:

$m_1 = \frac{4-2}{1-(-3)} = \frac{2}{4} = \frac{1}{2}$.

Using point-slope form with the point $(1, 4)$, we have

$y - 4 = \frac{1}{2}(x-1)$, so $y = \frac{1}{2}x - \frac{1}{2} + 4 = \frac{1}{2}x + \frac{7}{2}$.

For $x \ge 1$, the line passes through $(1, 4)$ and $(4, -2)$. The slope of the line is:

$m_2 = \frac{-2-4}{4-1} = \frac{-6}{3} = -2$.

Using point-slope form with the point $(1, 4)$, we have

$y - 4 = -2(x-1)$, so $y = -2x + 2 + 4 = -2x + 6$.

Therefore, the piecewise function is:

$f(x) = \begin{cases} \frac{1}{2}x + \frac{7}{2}, & x < 1 \\ -2x + 6, & x \ge 1 \end{cases}$

Now let's find the x- and y-intercepts.

For $x<1$, $y = \frac{1}{2}x + \frac{7}{2}$.

To find the y-intercept, let $x=0$. Then $y = \frac{1}{2}(0) + \frac{7}{2} = \frac{7}{2} = 3.5$.

To find the x-intercept, let $y=0$. Then $0 = \frac{1}{2}x + \frac{7}{2}$, so $\frac{1}{2}x = -\frac{7}{2}$, which means $x = -7$. Since $-7 < 1$, this is a valid x-intercept.

For $x \ge 1$, $y = -2x + 6$.

If $x=0$, $y = -2(0) + 6 = 6$. This is the y-intercept, but $0 \ge 1$ is false.

To find the x-intercept, let $y=0$. Then $0 = -2x + 6$, so $2x = 6$, which means $x = 3$. Since $3 \ge 1$, this is a valid x-intercept.

So the x-intercepts are $-7$ and $3$. The y-intercept is at $x=0$, which falls in the domain $x<1$, therefore, $y=3.5$.

3. Final Answer

The function notation is $f(x) = \begin{cases} \frac{1}{2}x + \frac{7}{2}, & x < 1 \\ -2x + 6, & x \ge 1 \end{cases}$.

x-intercepts: $-7$ and $3$.

y-intercept: $3.5$.