Problem 118 - College Algebra Essentials

Question

Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)

Step-by-Step Solution

Answer

To sketch the given piecewise function, one should draw a sloping line down from the left of the graph, stop and make a solid dot at the point (2,0). Then, from that point, draw a line that slopes upward to the right, extending to \(\infty\). This graph shows that the function decreases up till x=2, and then starts increasing for x>2, with a range from 0 to \(\infty\).

Step 1: Understand the Properties of the Function

Begin by considering each property individually. Knowing the function is decreasing on \((-\infty, 2)\) means the graph will slope downwards from left to right until it reaches x=2. Then on the interval \((2, \infty)\), the function is increasing, so the graph will slope upwards from x=2 onwards. The function value at x=2 is 0 which provides a specific point on the graph, namely (2,0). Finally, the range of the function is \([0, \infty)\), implying that the values of the function span from 0 to positive infinity.

Step 2: Sketch the Decreasing Part of the Function

For all values less than 2, the function is decreasing. Draw a line from somewhere above x=2 (because the function is decreasing and the value at x=2 is 0), going downwards from left to right and stopping at the point (2,0). Do not include the point (2,0) in this part of the line, as the function is not defined at x=2 in this interval.

Step 3: Sketch the Increasing Part of the Function

For all values greater than 2, the function is increasing. Starting from the point (2,0), draw a line going upwards from left to right. This line should extend forever in a positive direction, as the function continues increasing until \(\infty\).