Problem 118 - College Algebra

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

The graph starts at the x-axis at x = 2, sloping downwards to the left and upwards to the right, never crossing the x-axis.

Step 1: Understand the Decreasing Function

The function is decreasing on the interval \((- \infty, 2)\). This means, for all x-values in this interval, as x increases, f(x) decreases. Hence, draw a downward sloping line from left to x=2.

Step 2: Consider the Function Value at x=2

At x=2, the function value, f(2), equals 0. So, make sure the decreasing line from step 1 intersects the y-axis at 0 when x = 2.

Step 3: Understand the Increasing Function

The function is increasing on \((2, \infty)\). This means, for all x-values in this interval, as x increases, f(x) increases too. Therefore, draw an upward sloping line starting from where the previous line ended at x = 2.

Step 4: Consider the Range

The range of function is \([0, \infty)\), which means the y-values of the function will never go below 0. So, the graph drawn has to always stay on or above the y-axis.