Determine whether each relation defines y as a function of x. Giv...

Question

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5.

x = y⁶

Accepted Answer

Hello. Today we're going to be determining whether the relation between X and Y is a function, then we're going to find the domain and the range. So the relation given to us is X is equal to Y race to the fourth power. Now, if we use a graphic utility, this relation will be the shape of a Parabola that opens up to the right in the positive direction. Now, in order to determine whether this relation is a function or not, we're going to use the vertical line test. So we're going to start by drawing some vertical lines across the graph. And as long as each vertical line intercepts the graph at exactly one point that shows that the relation is a function. Now, if we take a look at the first vertical line, the vertical line intercepts the graph at two points, one in the positive region and one in the negative region. And the same can be said for the second vertical line. So since the vertical lines intercept the graph at exactly two points, that means the relation is not a function. So now that we know that the relation is not a function we're going to determine the domain and the range. Let's start with the domain. Now, the domain focuses on the lowest X value to the greatest X value in the graph. The vertex of the parabola starts at the origin. So that means the lowest X value is going to be zero. And since zero is a part of the part of the domain or is included on the graph, that means we're going to include the value of zero in our domain and we'll assign it a bracket. Next, we need to determine the highest value of X for the graph. Now, as the value of X increases, the graph is going to increase to positive infinity. What this means is that the greatest value of X is going to be infinity. So the domain is going to be from zero to infinity including the value of zero. Now, what we'll need to do is we'll need to identify the range. Now, the range focuses on the lowest Y value to the highest Y value. And currently, as XG becomes greater, the Y values will continue to decrease and increase to negative and positive infinity. That means that there is no limits to the Y values in both the negative and positive direction. Therefore, the range is going to be from negative infinity to positive infinity. So just to summarize, we use the vertical line test to determine that X is equal to Y to the fourth power is not a function, the domain is from zero to infinity including the value of zero. And the range is from negative infinity to positive infinity. With that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x = y⁶

Key Concepts

Function Definition

Domain and Range

Inverse Relations

Watch next