Answers:
- Absolute maximum: 4
- f(-3) = -2
- Absolute minimum: Does not exist
- Range:
- Domain:
- Is it a function? Yes
- Relative Maximum(s): 2
- Increasing Interval(s):
... interval notation
- Decreasing Interval(s):
- 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 indicating y can be anything between negative infinity and 4. We can include 4. The range in interval notation is
. 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 which condenses to the interval notation
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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 and also
. 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 and
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 . 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.