Question

NO LINKS!!!!! Please help me.

Absolute Maximum:
f(-3):
Absolute minimum:
Range:
Domain:
Is the graph a function?:
Relative Maximum(s):
Increasing Interval(s):
Decreasing Interval(s):
Relative minimum(s):​

THIS IS NOT MULTIPLE CHOICE!!!

Answer 1

Answers:

• Absolute maximum: 4
• f(-3) = -2
• Absolute minimum: Does not exist
• Range:
  `(-\infty, 4]`
• Domain:
  `[-4, \infty)`
• Is it a function? Yes
• Relative Maximum(s): 2
• Increasing Interval(s):
  `(-2, 1)` ... interval notation
• Decreasing Interval(s):
  `(-\infty, -2) \ \cup \ (1, \infty)`
• Relative Minimum(s): -4

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Explanations:

The absolute maximum occurs at the highest point. Specifically, it's the largest y output possible. In this case, it's y = 4.

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To determine the value of f(-3), we draw a vertical line through -3 on the x axis. Mark where this vertical line crosses the curve. Let's say its point P. From point P, draw a horizontal line until you reach the y axis. You should arrive at y = -2. Therefore, f(-3) = -2.

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The absolute min is similar to the absolute max, but now we're looking at the lowest y output possible. No such y value exists because the curve goes on forever downward.

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The range is the set of all possible y outputs. The range in compound inequality notation is
`-\infty < y \le 4` indicating y can be anything between negative infinity and 4. We can include 4. The range in interval notation is
`(-\infty, 4]`. Note the use of the square bracket so that we include the 4.

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The domain is the set of x inputs possible. The smallest such input allowed is x = -4. There is no largest input because the graph goes on forever to the right. The domain is any x value such that
`-4 \le x < \infty` which condenses to the interval notation
`[-4, \infty)`

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This is a function because we cannot draw a single vertical line through more than one point on this curve; hence, this graph passes the vertical line test.

Put another way, any x input in the domain leads to exactly one and only one y output. This is a nonvisual way to prove we have a function.

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A relative maximum occurs at any peak or mountain region. It is relatively the highest point in the neighborhood/region of points. There's one such mountain peak and it's at (1,2). We can think of this as a vertex of sorts for an upside down parabola. So the relative max is y = 2 because we're only concerned with the y value.

Note: y = 4 is not a relative max because there aren't any points to the left of that endpoint. A relative extrema must have points to the left and right of it for it to be a valid neighborhood.

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Imagine this curve represents a roller coaster. As we move to the right, going uphill on this curve is an increasing section. That would be the interval from x = -2 to x = 1. So we'd say -2 < x < 1 which condenses to the interval notation (-2, 1). This is not to be confused with ordered pair (x,y) notation.

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Now we consider when we move downhill when we move to the right. This occurs on the intervals
`-\infty < x < -2` and also
`1 < x < \infty`. We don't include any of the endpoints. This is because at x = -2 and x = 1, the cart is neither moving uphill nor downhill. These locations are stationary resting points so to speak.

Those two inequalities mentioned convert to the interval notations
`(-\infty, -2)` and
`(1, \infty)` in that order.

Once we determined those separated disjoint regions, we glue them together with the use of the union symbol U.

Our answer for this part would be
`(-\infty, -2) \ \cup \ (1, \infty)`. Any point in this collective region will be moving downhill when moving left to right.

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This is similar to a relative maximum, but this time we're looking at the lowest valley point of a certain neighborhood. This is at (-2,-4). Therefore, the relative min is y = -4.