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Chapter 13: Problem 79

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-4(y-1)^{2}+3$$

Short Answer

Expert verified
The domain of the function is (-鈭�, 3] and the range is (-鈭�, 鈭�). The given relation is not a function.

Step by step solution

01

Identify the Vertex

The vertex of a parabolic equation is given by the point (h,k). In the given equation \(x=-4(y-1)^{2}+3\), the vertex is (3,1).
02

Calculate the Domain

Since the parabola opens to the left, the x values will extend to the point of the vertex from negative infinity. Hence the domain of the function is (-鈭�, 3].
03

Calculate the Range

As the parabola opens to the left and right, the y values extend from negative infinity to positive infinity. So, the range is (-鈭�, 鈭�).
04

Check if the relation is a function

In this context, a relation is a function if for each x there is exactly one y in the function. As per the vertical line test, we can see that a vertical line cuts the parabola in more than one place. This means that for a certain x value, there exists more than one y value. Hence, the given relation is not a function.

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