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Chapter 13: Problem 80

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

Short Answer

Expert verified
The vertex of the parabola is (1, -2) and it opens to the left. The domain is \((-\infty, -2]\) and the range is \((-\infty, \infty)\). The relation is indeed a function.

Step by step solution

01

Identify the vertex and direction of the parabola

The given equation is in the vertex form \(x=a(y-h)^2+k\). From this equation, \(a=-3\), \(h=1\) and \(k=-2\). Thus, the vertex of the parabola is \((h, k) = (1, -2)\). Since \(a=-3\), which is less than 0, the parabola opens to the left.
02

Determine the domain and range of the relation

Because the parabola opens to the left, the x-values decrease as y moves up or down from the vertex. This means the domain of the relation, which is the set of all possible x-values, is \((-\infty, -2]\). The parabola continues indefinitely upward and downward, so the range of the relation, i.e., the set of all possible y-values, is \((-\infty, \infty)\).
03

Check if the relation is a function

A relation is a function if each input has exactly one output. In this case, if we choose a specific y-value, there's exactly one corresponding x-value from the equation. Therefore, the relation is indeed a function.

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