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.

y = √4x + 1

Accepted Answer

Hello. Today we're going to be determining whether the given relation between X and Y is a function or not, then we'll be finding the domain and the range. So we are given Y is equal to the square root of five X plus 14. Now, if we use a graphene utility, the plot of the graph is going to start at the X value negative 2.8 then the graph is going to increase to the right. So let's go ahead and first determine whether this is a function or not. Now, in order to determine whether this is a function, we can go ahead and use the vertical line test, we're going to start by drawing a few vertical lines across the graph. And as long as the vertical line intercepts the graph at exactly one point that will show that this graph is a function. So if we take a look at the first vertical line, the first vertical line intercepts the graph at exactly one point, the same can be said for the second and the third vertical line. So since each of the vertical lines intercepts the graph at exactly one point, this shows that the given relation is a function. Now, what we wanna do is we want to find the domain of the given function. Now, Y equal to the square root of five X plus 14 is a square root equation. If we have a general square root equation such as Y is equal to the square root of X, the value X must be greater than or equal to zero. Having a negative number inside of the square root will produce an un an unvalidated answer. So in order to determine the domain of the given function, we're going to have to take the insight argument which is five X plus and set it greater than or equal to zero. Then what we'll need to do is we'll need to solve for the values of X. So in order to solve for X in the inequality, we're going to have to subtract 14 to both sides of the equation that will leave us with five X greater than or equal to negative 14. The next thing we'll need to do is we'll need to divide both sides of the inequality by five that will leave us with X is greater than equal to negative 14/5. The value negative 14/5. When plugged into a calculator will give us an approximate value of negative 2.8. What this means is that the domain or the values of X must be greater than or equal to negative 2.8. Now, the domain focuses on the smallest value of X to the largest value of X. Since X must be greater than or equal to negative 2. that means the smallest value of X must be negative 2.8. And we're going to be including that value on the graph. So the domain is going to start with the value negative 2.8. And since it is included, we're going to assign it a bracket. Next, we're going to determine the largest value of X for the domain. Now, the largest value of X, if we take a look at the graph is going to be positive infinity as the graph increases along the X axis is is going to continue into positive infinity. So the domain is going to end with positive infinity, the domain will be negative 2.8 to positive infinity. Now, what we're going to do is we're going to determine the range of the graph. Now, the range focuses on the smallest Y value to the largest Y value. No, we have an X intercept at negative 2.8 but this X intercept lies at the value Y is equal to zero. So that means the smallest value of Y for the range is going to be zero. And since the point at negative 2.8 is included in the graph, we're going to be including the value of zero. Now we need to determine the largest value of Y will notice that as this graph continues along the X axis, this graph is gonna slowly increase over time. What this means is that the value of Y is going to approach positive infinity over time. And that means the range is going to be from zero to positive infinity. So just to summarize, we use the vertical line test to determine that Y is equal to the square of five X plus 14 is a function. The domain is going to be negative 2.8 included to positive infinity. And the range is going to be from zero included to positive infinity. And with that being said, the answer to this problem is going to be C 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. y = √4x + 1

Key Concepts

Function Definition

Domain and Range

Square Root Function

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