Problem 54 - College Algebra

Question

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

Step-by-Step Solution

Answer

The vertex is at (-2, 1); the parabola opens to the left. The domain is \((- \infty, -2]\) and the range is \((- \infty, \infty)\). The relation is not a function.

Step 1: Identifying Vertex and Direction of Opening

The parabola equation is given in vertex form as \(x=a(y-k)^{2}+h\). If a is less than 0, the parabola opens to the left. If a is greater than 0, the parabola opens to the right. Here, a is equal to -3, which is less than 0, so the parabola opens to the left. Vertex is at point(h, k). In our equation, h is -2 and k is 1, which gives us the vertex at (-2, 1).

Step 2: Determining the Domain

For a parabola that opens to the left or right, the domain is an interval. Since our parabola opens to the left (meaning there is a maximum x-value), the domain is \((- \infty, h]\), in other words, \((- \infty, -2]\).

Step 3: Determining the Range

The range of a parabola that opens to the left or right is all real numbers because there are no minimum or maximum y-values. Therefore, the range is \((- \infty, \infty)\).

Step 4: Is it a Function?

A relation is a function if every input has exactly one output. In this equation, for every x in the domain, there can be two corresponding values of y as the graph is a downward parabola. Therefore, it is not a function.