Questions: The function (f) is defined as follows. [ f(x)=leftbeginarrayll -x+3 text if x<1 3 x-1 text if x geq 1 endarrayright. ] (a) Find the domain of the function. (b) Locate any intercepts. (c) Graph the function. (d) Based on the graph, find the range. (a) The domain of the function (f) is (square) ]. (Type your answer in interval notation.) (b) Locate any intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The intercept(s) is/are (square) ]. (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no intercepts. (c) Choose the correct graph below. A. B. C. D.

Solution Steps

The function is defined for all x-values less than 1 and for all x-values greater than or equal to 1. Combining these, the function is defined for all real numbers.

• x-intercept: The x-intercept occurs when f(x) = 0.

  For -x + 3 = 0 (when x < 1), we get x = 3. Since 3 is not less than 1, this is not a valid x-intercept.

  For 3x - 1 = 0 (when x ≥ 1), we get x = 1/3. Since 1/3 is not greater than or equal to 1, this is not a valid intercept. Therefore, there are no x-intercepts.

• y-intercept: The y-intercept occurs when x = 0. Since 0 < 1, we use the first piece of the function: f(0) = -0 + 3 = 3. So the y-intercept is (0, 3).

The graph will consist of two parts:

1. For x < 1, the graph is the line y = -x + 3. This line has a slope of -1 and a y-intercept of 3. Since it's only defined for x < 1, there will be an open circle at the point where x = 1.

2. For x ≥ 1, the graph is the line y = 3x - 1. This line has a slope of 3 and would have a y-intercept of -1 if extended, but it starts at x = 1 with a closed circle.

The correct graph will show these two line segments, with the first one ending with an open circle at x = 1 and the second starting with a closed circle at x = 1. This corresponds to graph A.

Final Answer