Question

express the function graphed on the axes below as a piecewise function.

Explanation:

Step1: Analyze the left segment

The left part of the graph is a horizontal line at \( y = 4 \) with an open circle at \( x=-1 \), so it applies for \( x < -1 \). The function here is \( f(x)=4 \).

Step2: Analyze the right segment

The right part is a line. Let's find its slope. The open circle is at \( (3,6) \), and let's take another point, say \( (4,7) \) (since it's a line with slope 1, as the change in y is 1 when change in x is 1). The equation of the line: using point - slope form \( y - y_1=m(x - x_1) \), where \( m = 1 \), \( x_1 = 3 \), \( y_1 = 6 \). So \( y-6=1\times(x - 3)\), which simplifies to \( y=x + 3 \). The domain for this part is \( x>3 \) (since the open circle is at \( x = 3 \)).

Answer:

\( f(x)=

$$\begin{cases}4& \text{for } x < - 1\\x + 3& \text{for } x>3\end{cases}$$

\)