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.

ƒ(x) = |2x + 1|

Accepted Answer

Welcome back. Everyone. In this problem, we want to identify the intervals where the function H of X equals the absolute value of X plus four is 1 to 1 and thus has an inverse A says it's between open bracket four infinity, closed parentheses. B open parentheses, negative infinity, negative four closed bracket and closed, sorry, open bracket. Four infinity, closed parentheses. C says open parentheses, negative infinity, negative four closed bracket and open bracket, negative four infinity closed parentheses. And D says open parentheses, negative infinity, foreclosed bracket and open bracket four infinity closed parentheses. Now, how are we going to figure out these intervals for our function of X? OK. Well, a sketch of its graph would really help. So let's just draw our axes here. OK. And this, we have of X on our vertical axis and X on our horizontal axis. And now we want to figure out where the function is 1 to 1. Now, what do we know about this function? It's an absolute value. Well, we know that for H of X equals X plus four, its behavior changes at a point. Now when X is zero H of X would be four. And when X is negative 4h of X would be equal to zero. OK? But the thing is if we go below negative four, OK. For example, if we go to say uh negative eight, then H of X would be negative eight plus four, which equals negative four. But we're finding the absolute value. So it would actually be four. OK. So our graph would look something like this. Let me try to draw a good sketch here. So a graph would look something like this. OK? And its behavior changes like we said at the point where X equals negative four. Now, in this case, OK, that tells us then that where the function is 1 to 1 that is where there is one input for each output would be in this shaded part of our graph. OK? And the other shaded part of our graph, let's let me share that in red here. OK. Now what are those intervals going to be? Well notice in the positive direction, OK. That interval it starts at negative four. OK. Sorry. Yes, it starts at negative four. It includes negative four and it goes all the way to infinity. OK. So that would be one of the intervals for the next interval. Notice that it starts at negative four going towards the left, but it also includes negative four. OK. So it goes to negative infinity, open parentheses and comes all the way to negative four, close the bracket. Therefore, these are the intervals where our function is 1 to 1 and thus has an inverse. If we look back at our answer choices, that means C 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.

ƒ(x) = |2x + 1|

Key Concepts

Inverse Functions

Horizontal Line Test

Piecewise Functions

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