How do you find the domain and range and determine whether the relation is a function given :#x=2y^23#?
See below.
For y as a function of x.
Rearrange;
Add 3 to both sides:
Divide by 2:
Taking roots of both sides:
For positive root:
For the domain we must make sure the numerator is
The numerator is
So the domain is
The range for the positive root is:
As
So the range is:
For negative root the domain is the same for real numbers. The range is in the reverse direction:
The determination of a function.
For positive root only a function is defined.
For negative root only a function is defined.
For
This is not a function. A function is only defined if an element in a domain is mapped onto one and one only element in a range. In the case of both roots this would map one element in the domain to two elements in the range. This is sometimes called a one to many relationship.
See graph:
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To find the domain and range of the relation x = 2y^2  3 and determine if it is a function:

Domain: The domain of the relation is all possible values of y that will produce a real number for x. In this case, there are no restrictions on y, so the domain is all real numbers.

Range: To find the range, we need to determine all possible values of x that the relation can produce. Since the equation is quadratic, it opens upwards (positive coefficient of y^2), and the lowest point occurs when y = 0. Substituting y = 0 into the equation, we get x = 2(0)^2  3 = 3. Therefore, the range is all real numbers greater than or equal to 3.

Function: To determine if the relation is a function, we need to check if each input value of y corresponds to exactly one output value of x. Since the equation is quadratic, for each value of y, there are two possible values of x (one positive and one negative), so the relation is not a function.
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To find the domain and range and determine whether the relation (x = 2y^2  3) is a function:

Domain: The domain of the relation refers to all possible values of (y) for which (x) is defined. Since (x) is defined for all real numbers, the domain is all real numbers.

Range: The range of the relation refers to all possible values of (x) resulting from the chosen values of (y). To find the range, we need to solve the equation for (y) in terms of (x):
[ x = 2y^2  3 ]
[ x + 3 = 2y^2 ]
[ \frac{x + 3}{2} = y^2 ]
[ y = \pm \sqrt{\frac{x + 3}{2}} ]
Since (y) can take both positive and negative square roots, the range is all real numbers.
 Function: To determine if the relation is a function, we must ensure that each input (y) corresponds to exactly one output (x). If there are multiple values of (x) for a single value of (y), the relation is not a function.
In this case, since (y) can take both positive and negative square roots, for each (y), there are two possible values of (x). Therefore, the relation is not a function.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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