Increasing and Decreasing Functions

Question

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = 1/|x|

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to analyze and graph the function Y is equal to -3 divided by the absolute value of X. What symmetries does it exhibit and during which intervals is it increasing or decreasing? Below the problem we're given an empty graph on which to plot our function. Now, the first thing we want to check for are symmetries, because that's gonna allow us to um graph our function a little bit easier if we do have even or odd symmetries. So, to check for even an odd symmetries, we're going to look at Y of minus X. So, we're basically going to replace X with minus X in our function. So this is going to be equal to -3, divided by the absolute value of minus X. Which is equal to minus 3 divided by the absolute value of X, since the absolute value function can get rid of the negative sign. But this is just equal to our original function, Y as a function of X. So that means that we actually have even symmetry. So that is the first part of the question answered. And since we have even symmetry, that means that our function is symmetric about the Y axis. So next we need to determine the intervals over which our function is increasing or decreasing. So we're gonna look at first, the case of when X is greater than 0. So if we look at this case, and we look at our function Y is equal to -3 divided by the absolute value of X, for small values of X, we're going to have minus 3 divided by a small number, which is going to give us a very large negative value. But as X gets larger, the minus 3 divided by the absolute value of X is going to approach a smaller negative value. So we're starting from a large negative value going to a smaller negative value. That means that our function is going to be increasing on this interval. So our function is increasing. On the open interval from 0 to infinity. So that is one of our intervals found. Next, we're going to look at X, less than 0. And in this case, if we have a very large negative value for X, the absolute value. Of X makes that a positive quantity. And then we'll have -3 divided by a very large positive value, which is going to be very close to 0, so we'll have a small negative value. But as we approach X equal to 0, our denominator in a fraction is going to become very large. So we'll have minus 3 divided by some small positive value. And so that means we're gonna get an overall large negative value for Y. So we're going from a small negative value for Y. To a large negative value Y, so that means that our function is going to be decreasing on this interval. So, for x less than 0, our function is decreasing. And if we want to write that in interval notation, we have decreasing on the open interval from minus infinity to 0. That is our 2nd interval. That we have found And so now we just need to use this information to plot our function. So to do this, we're going to determine a couple of points. So first we're gonna look at the case when X is equal to 1, and if we evaluate why when X is equal to 1, we get minus 3. And so we'll plot that point on our graph, and by symmetry, that means that X equal to -1 also has a Y value of -3. Next, we'll look at the case when X is equal to 3. In that case, our Y value is going to be equal to -1, so we'll plot that point on the graph, and by symmetry, X equal to -3 has that same Y value. So now we just need to look at the behavior as X approaches plus or minus infinity. So as X approaches positive infinity, we're going to have -3 divided by a very large number. So we're going to be approaching the value of 0. And the same is true when X is approaching minus infinity. Since we have the absolute value of X, we're still going to have -3 divided by a very large number, which is approaching 0. And as X approaches 0, We're going to get a very large negative number, since we'll have minus 3 divided by A very small positive quantity, which will give us an overall negative number. So we're approaching minus infinity as X approaches 0 from both sides. So, we're going to use that information to actually draw our graph. So on the left hand side, we're gonna start off, coming from minus infinity, it's gonna be close to 0. We're going to Then cross the two points that we have plotted at X equal to minus 3. And X equal to. -1. And we're going to continue to approach the y axis, but as we do so, we're going to approach a very large negative value. And so now we're just going to mirror this on the right hand side of the y axis since we have even symmetry. But here we're actually increasing. I want to start off from a very large negative value to the point where x equal to 1 we'll cross the point where x equals to 3 as well, and then continue our graph to approach the x axis. So Y equaling approaching 0. And so this is a graph that the question was looking for. It looks similar to a Y shape. So I hope that this explanation has been useful, and I'll see you in the next video.

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = 1/|x|

Key Concepts

Increasing and Decreasing Functions

Graphing Functions

Symmetry in Functions

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