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=y^4

Accepted Answer

Everyone in this problem were asked to find out if the given equation represents a function and also to find the domain in range. The equation we're given is 81 X is equal to Y to the exponent four divided by 16. We're given four answer choices A through D and we're gonna come back to those as we work through this problem. Now, if we want to figure out if this is a function, what we wanna do is look at every single X value and make sure it has only one corresponding Y value. No. Yeah. Oops 81 X is equal to Y to the exponent four divided by 16. And let's think about an example where we have X is equal to one divided by 16. If we have X is equal to one divided by 16, we get that 81 divided by 16 is equal to Y to the exponent four divided by 16. It can multiply both sides by 16. Calls us at 81 is equal to Y to the exponent four. And if we take the four through of 81 we get that Y is equal to positive or negative three, OK? We can substitute Y is equal to positive or negative three, raise it to the exponent four and we'll get 81. And so what this shows us is that some X values have more than one corresponding Y value. And so this is not a function. And you can imagine if you were to do a vertical line test if you had the graph of this function and you choose a particular X value, you draw a vertical line, OK? There are gonna be two Y values associated with that XX value. And so your vertical line is gonna pass through two points on the graph which indicates that it's not a function. So in this case, we have not a function and we can eliminate options C and D from our answer choices because they both said that this was a function. All right. So we know this is not a function. Now let's move to the domain and the range now for the domain this is gonna be all possible X values. OK? So we can substitute into this equation so that our equation is defined looking at our equation on the right hand side Y to the exponent four, that's always gonna be positive and that's an even exponent. So no matter what our Y value is, that's gonna be positive, we divide it by 16, which is positive. So the right hand side of our equation will always be positive, which means that the left hand side must also be positive in order for it to be equal. OK? So we need X therefore to be positive. If X is negative, then we're taking the fourth route of a negative value which is not possible and we do not get real values doing that. And so we need X to be greater than zero and it can also be equal to zero. If X is equal to zero, the left-hand side is equal to zero and then Y is just going to be equal to zero as well. All right. So we have our domain from zero to infinity. We use a square bracket on the left hand side to indicate that we include zero. And then we have our range. And what's our range gonna be? Well, our range is actually gonna be any value from negative infinity all the way up to positive infinity on the left, we're gonna have a, any possible positive value when we take the fourth route. Remember just like in our example, we get the positive and the negative root. And so our range is gonna be all possible values. So if we look at our answer choices now they had already eliminated C N D and the correct answer is going to be option A option B does not have the restriction on the domain that we need. So the correct answer is a, this is not a function, the domain goes from zero. To infinity including zero and the range is all values from negative infinity to positive infinity. Thanks everyone for watching. I hope this video helped see you in the next one.

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x=y^4

Key Concepts

Function Definition

Domain and Range

Inverse Relations

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