Use the following graph of a function f to tell what intervals the function is increasing or decreasing on.

On the interval (−∞,
1

3
] the function is Increasing .
On the interval [
1

3
,1] the function is Decreasing .

Question

Use the following graph of a function f to tell what intervals the function is increasing or decreasing on.

On the interval (−∞,
1

3
] the function is Increasing .
On the interval [
1

3
,1] the function is Decreasing .

BlackTom AI · Accepted Answer

The question presents a piecewise description of where the function f is increasing or decreasing based on a graph, though the interval endpoints in the prompt appear garbled. The task is to assign, for each specified interval, whether f is Increasing or Decreasing.

First, consider the left interval indicated: (−∞, …). When a graph is increasing on a leftmost interval, it means as x moves to the right within that interval, f(x) grows. If the graph shows the function rising as we move from far left up to the first turning point, then the correct label for this interval would be Increasing. In this prompt, the suggested label in the answer field for this left interval is Increasing, which aligns with typical behavior on a leftward interval before a peak.

Next, turn to the right interval indicated: …,1] (the exact right endpoint is unclear due to formatting, but it is described as a subsequent interval after the left one). If the graph shows the function falling as x increases across this right-hand interval (i.e., the portion after the peak where the slope is negative), then f is Decreasing on that interval. The provided answer option for this interval is Decreasing, which matches the usual right-side behavior after a local maximum.

Important potential sources of confusion to watch for:
- Ambiguity in endpoints: The garbled formatting makes the exact numeric endpoints unclear, but the qualitative behavior—left interval increasing, right interval decreasing—fits a typical single-peak scenario.
- Interval order: Ensure you match the first listed interval with the first dropdown (Increasing) and the second listed interval with the second dropdown (Decreasing). A mismatch would indicate a misreading of the graph rather than a property of f.
- Graph-based interpretation: This reasoning depends entirely on the slope indicated by the graph on each interval. If the graph actually rose to a peak and then fell, the described labels are correct; if the graph behaved differently (e.g., decreasing then increasing), the labels would flip accordingly.

In summary, given the conventional interpretation of a left interval where f rises and a subsequent interval where f falls, labeling the first interval as Increasing and the second as Decreasing is consistent with a typical unimodal, single-peak function on those portions.

MAT135H5 S LEC0102 1.1 Preparation Check

Use the following graph of a function f to tell what intervals the function is increasing or decreasing on. On the interval (−∞, 1 3 ] the function is Increasing . On the interval [ 1 3 ,1] the function is Decreasing .

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