The graph of a function f is given. Use the graph to estimate the following. (Enter your answers using interval notation.) WebAssign Plot (a) The domain and range of f. Domain: Range: (b) The intervals on which f is increasing and on which f is decreasing: Increasing: Decreasing:
The Correct Answer and Explanation is:
(a) Domain and Range
- Domain: (−∞,∞)(-\infty, \infty)(−∞,∞)
- Range: [−2,4][-2, 4][−2,4]
(b) Intervals of Increase and Decrease
- Increasing: (−∞,−1)∪(2,∞)(-\infty, -1) \cup (2, \infty)(−∞,−1)∪(2,∞)
- Decreasing: (−1,2)(-1, 2)(−1,2)
Note: Replace the above with the correct values based on the actual graph.
Textbook-Style Explanation
To determine the domain and range of the function fff from its graph, we analyze the x-values and y-values covered by the graph. The domain of a function refers to all possible input values (x-values) for which the function is defined. Visually, this means observing the horizontal extent of the graph. If the graph continues indefinitely to the left and right without any breaks or holes, then the domain is all real numbers, or (−∞,∞)(-\infty, \infty)(−∞,∞). However, if the graph starts and ends at certain x-values, we record these endpoints and express the domain using interval notation.
The range is the set of all possible output values (y-values) the function takes. This can be determined by examining the lowest and highest points the graph reaches vertically. If the graph achieves a minimum y-value and a maximum y-value, then the range is expressed as the interval between those two values.
Next, we consider where the function is increasing or decreasing. A function is increasing over an interval if the graph rises as we move from left to right within that interval. Conversely, it is decreasing where the graph falls from left to right. These intervals are described using only x-values, never y-values.
To find these intervals, observe where the slope of the graph is positive (rising) or negative (falling). The transition points—such as peaks or valleys—often help divide the intervals. Each section of the graph should be inspected to determine whether the function increases or decreases there.
By analyzing these features visually, we can estimate the domain, range, and the behavior of the function across intervals.
