Where do inverses exist? Use analytical and/o...

Question

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} ƒ(x) = 1/(x-5)

Accepted Answer

Welcome back. Everyone. In this problem, we want to identify the intervals where the function H of X equals two divided by three, X is 1 to 1 and thus has an inverse A says it's between open parentheses, negative infinity, three closed bracket and open bracket. Three infinity, closed parentheses. B says it's between open parentheses, negative infinity, three closed bracket, C says open parentheses, negative infinity, negative three closed parentheses and open parentheses, three infinity, closed parentheses and D open parentheses, negative infinity, negative three closed parentheses and open parentheses, negative three infinity closed parentheses. Now, if we're going to identify the intervals where our function is 1 to 1, OK. Let's think about the nature of our function for a bit. OK. So let's take a good look at our function of X equals two divided by three plus X. Now, we know that our function is asymptotic. OK. And it has a vertical Asymptote. But what do we know? Or at what point will our function be undefined? Well, remember a function is undefined if its denominator is equal to zero. OK. And that means then at the point where H of X is undefined, OK. Then it's denominator three plus X. Let me write it below it here plus X is going to be equal to zero. Thus, that means then that at that point X must be equal to negative three. So what does this mean? Well, if we were to plot our graph of H of X, let's just do a quick sketch here. It means that it will have a function, it's gonna have a function for all of the points after X being equal to negative three and all of the points before X being equal to negative three. OK? Where our vertical axis represents H of X and our horizontal axis represents X but no, at the point where X equals negative three, OK. Then our function is undefined. Therefore, it's 1 to 14, both intervals going in the positive and negative X direction except when X equals negative three. Now, what are those intervals going to be? Well here notice that not including negative three in the left direction it would go from, let me, let me write it here going in the going to the left, sorry, it would start at negative three and go all the way to negative infinity. And it wouldn't include both of those values. So we use an open parenthesis to represent that next. Going to the right, it would start at negative three, not including negative three and go all the way to infinity. Therefore, for the intervals, open parentheses, negative infinity, negative three closed parentheses and open parentheses, negative three infinity closed parentheses H of X is 1 to 1 and has an inverse. Therefore, D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} ƒ(x) = 1/(x-5)

Key Concepts

Inverse Functions

Domain and Range

Horizontal Line Test

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