Answers:
- Domain: [tex]\big[-9, \infty\big)[/tex]
- Range: [tex]\big(-\infty, 3\big][/tex]
- Absolute Max: y = 3
- Absolute Min: None
- Relative Max: y = 3
- Relative Min: None
- x-intercepts: x = -6 and x = 6
- y-intercept: y = 3
- Increasing: on the interval (-9, -3)
- Decreasing: on the interval [tex](4, \infty)[/tex]
- Constant: on the interval (-3, 4)
- f(8) = -3
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Explanation:
The domain is the set of allowed x inputs.
The left-most part of the graph is at x = -9. There is no right-most part since it goes on forever in that direction due to the arrow.
So we can say [tex]x \ge -9[/tex] or [tex]-9 \le x[/tex] or [tex]-9 \le x < \infty[/tex]
That third inequality mentioned then turns into the interval notation [tex]\big[-9, \infty\big)[/tex]
Use a square bracket to include -9 as part of the domain. It's included because of the closed filled in circle, in contrast to a open hole.
We always will use a curved parenthesis for either infinity since we cannot actually reach it (it's not a number to reach).
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Now onto the range. This is the set of possible y outputs.
The graph stretches forever downward, so there is no lower bound. The upper bound, or highest part, is when y = 3
The range as a compound inequality is [tex]-\infty < y \le 3[/tex] which condenses to the interval notation of [tex]\big( -\infty , 3 \big][/tex]
Once again, use a square bracket to include the endpoint 3.
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The absolute max is the highest point of the graph. It's the peak of the mountain. In this case, the absolute max is y = 3. This is the relative max also. The relative max is the maximum within a certain specified interval.
There is no absolute min since the graph goes downward forever. There is no lowest point. This also rules out a relative min as well.
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The function is increasing when it goes uphill as you move to the right.
The graph shows the function increases on the interval -9 < x < -3 which condenses to the interval notation (-9, -3). This is NOT ordered pair notation which unfortunately looks identical to interval notation when using curved parenthesis for both endpoints.
In contrast, the function decreases when it goes downhill (when moving to the right). This occurs when [tex]4 < x < \infty[/tex] which shortens down to the interval notation of [tex](4, \infty)[/tex]
The graph is flat or constant when it's neither increasing nor decreasing. The graph is constant when -3 < x < 4 which translates to the interval notation of (-3, 4)
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Now onto the x and y intercepts
The x intercepts are where the graph crosses the horizontal x axis.
We have the two locations of x = -6 and x = 6 as the x intercepts
The y intercept is at y = 3, which is where it crosses the y axis. This also happens to be the max value mentioned earlier, though the y intercept won't always be the max.
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Lastly, let's compute f(8)
Locate 8 on the x axis. Then draw a vertical line through this. The vertical line crosses the curve at (8, -3). Then move to the left until hitting y = -3 on the y axis
So x = 8 plugged in leads to y = -3 as the output.
Therefore, f(8) = -3
