Math Problem Statement

Give me the increase intervals , decreasing intervals , positive interval,negative intervals,X-intercept ,y intercept , average range , domain with set notation

Solution

Based on the image provided, I'll extract the details for the key features of the function as requested. The function appears to be a piecewise linear function depicted on a graph. Here's the information extracted:

1. Increasing Intervals:

The function is increasing where the graph slopes upwards.

  • Interval: [2,6][2, 6]

2. Decreasing Intervals:

The function is decreasing where the graph slopes downwards.

  • Interval: [2,2][-2, 2]

3. Positive Intervals:

The function is positive where the graph is above the x-axis.

  • Interval: (0,6](0, 6]

4. Negative Intervals:

The function is negative where the graph is below the x-axis.

  • Interval: [2,0)[-2, 0)

5. X-Intercepts:

The x-values where the graph crosses the x-axis.

  • X-intercepts: (0,0)(0, 0) and (6,0)(6, 0)

6. Y-Intercept:

The y-value where the graph crosses the y-axis.

  • Y-intercept: (0,0)(0, 0)

7. Domain (Set Notation):

All possible x-values for which the function is defined.

  • Domain: [2,6][-2, 6]

8. Range (Set Notation):

All possible y-values that the function can take.

  • Range: [3,3][-3, 3]

9. Average Rate of Change:

The average rate of change over an interval of the function is a measure of how much the function's output (y-value) changes per unit change in the input (x-value).

  • Interval: [4,6][4, 6]
  • Average Rate of Change: 0364=1.5\frac{0 - 3}{6 - 4} = -1.5

Would you like further details or have any questions? Here are some related questions you might find interesting:

  1. How do you determine the intervals of increase and decrease from a graph?
  2. What does the average rate of change tell you about a function?
  3. How does the domain and range of a function impact its graph?
  4. Can a function have more than one x-intercept and y-intercept?
  5. What is the significance of the positive and negative intervals of a function?

Tip: When analyzing a function graph, always pay close attention to where the graph crosses the axes, as these points provide critical information about the function's intercepts and intervals.