Questions: 3.2 Homework 3.2 homework Sketch the graph of the piecewise function and write the domain in interval notation. f(x) = x-3 if x < -1 -3x + 1 if x ≥ -1 Hint: Draw each piece as a ray with the starting point at the edge of the domain. Then draw an open or solid dot at the starting point of each ray. If two rays have the same starting point, only draw one dot.

Sketch the graph of the piecewise function and write the domain in interval notation. \[ f(x)=\left\{\begin{array}{ll} x-3 & \text { if } x<-1 \\ -3 x+1 & \text { if } x \geq-1 \end{array}\right. \] Graph \(f(x) = x - 3\) for \(x < -1\).

Choose two \(x\) values less than \(-1\), say \(-3\) and \(-2\). When \(x = -3\), \(f(-3) = -3 - 3 = -6\). The point is \((-3, -6)\). When \(x = -2\), \(f(-2) = -2 - 3 = -5\). The point is \((-2, -5)\). Plot the points and draw a ray that starts at \((-1, -4)\) with an open circle and passes through the two points. Graph \(f(x) = -3x + 1\) for \(x \geq -1\).

Choose two \(x\) values greater than or equal to \(-1\), say \(-1\) and \(0\). When \(x = -1\), \(f(-1) = -3(-1) + 1 = 4\). The point is \((-1, 4)\). When \(x = 0\), \(f(0) = -3(0) + 1 = 1\). The point is \((0, 1)\). Plot the points and draw a ray that starts at \((-1, 4)\) with a closed circle and passes through the point \((0, 1)\). Write the domain in interval notation.

The first piece is defined for \(x < -1\) and the second piece is defined for \(x \geq -1\). Combining these, we get the domain of all real numbers. Domain: \((-\infty, \infty)\)

The graph is shown below with an open circle at \((-1,-4)\) and a closed circle at \((-1,4)\). The domain of the piecewise function is \((-\infty, \infty)\).

The graph consists of two rays, one starting at \((-1, -4)\) with an open circle and extending to the left, and the other starting at \((-1, 4)\) with a closed circle and extending to the right. The domain of the piecewise function is \(\boxed{(-\infty, \infty)}\).