Theory and Examples

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Question

Theory and Examples

[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.

h(x) = |x + 2| − |x − 3|, −∞ < x < ∞

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to graphic function determine its absolute extreme values, and we're given the function F of X, which is equal to the absolute value of the quantity of X plus 1 in quantity minus the absolute value of the quantity of X minus 5 in quantity. So the problem will get an anti graph in which to plot our function. Now, if we look at a function FX, we actually have two absolute value functions being subtracted from each other. And recall that the absolute value function actually changes behavior whenever the quantity inside that absolute value is equal to 0. So this is going to occur when X + 1 is equal to 0, which means X is equal to -1, and also when X minus 5 is equal to 0, or when X is equal to 5. Now we just need to determine the behavior of a function FX between these two points. So, first we're gonna look at the case when X is less than -1. Now, in this case, both quantities, X plus 1 as well as X minus 5, are both going to be negative values. So in order to get rid of the absolute value sign in our function F of X, we'll need to write F of X as being equal to minus the quantity of X + 1 in quantity, minus the quantity of minus the quantity of X minus 5 in quantity. And when we simplify this expression, this is going to be equal to -6. So function F of X is just a constant of -6 when X is less than -1. Now we need to look at the case when -1 is less than X, which is less than 5. Now, in this case, The quantity of X + 1 is actually going to be positive, so we can just remove its absolute value signs. So we'll have F of X. is equal to x + 1. Minus the quantity, but here, the quantity of X minus 5 is still going to be negative, so to get rid of the absolute value sign, we'll have to have minus the quantity of X minus 5 in quantity. And simplifying this expression, this is going to be equal to. 2 X minus 4. So, in between these two points, our function FX is going to follow a line that has the equation of 2X minus 4. And then finally, if we look at the case when X is greater than 5, In this case, both quantities of X + 1 and X minus 5 are both positive quantities. So, to get rid of the absolute value signs, we just need to remove those. So we'll have F of X. is going to be equal to x + 1 minus quantity of X minus 5 in quantity. And simplifying this expression, this is going to be equal to 6. So function F of X is a constant with a value of 6. 4 X values greater than 5. So now what we're going to do is plot. These two points on our graph still have when X is -1. F of -1 is equal to -6. And when we look at X equal to 5, We see that our function F of 5 is equal to 6. So now we just need to connect these points with straight lines. So, we'll have a horizontal line coming in from X equal to minus infinity, to the point where we have X equal to -1. Then we'll need to connect the points at -16 to the point of 5.6, and that will give us a line that follows the equation of 2 X minus 4. Then finally, we'll have a horizon line. With a Y value of 6, starting with X. Equal to 5 and going towards X equal to infinity. So, the graph that we have drawn on the screen here is part of the answer to this problem. So next we need to determine the absolute extreme values, and for this, we'll just need to analyze our graph here. Now, if we look for the maximum value, That our function F of X takes, we see that we have for our absolute maximum. A value of 6. And this occurs on the right open interval. Of 5 to infinity. And if we look for the smallest value that our function F of X takes, that will give us the absolute minimum. And we actually have an absolute minimum of -6, and this actually occurs on the left open interval from minus infinity. 2 minus 1. And so those are the answers that this problem was looking for. So I hope that this explanation has been useful, and I'll see you in the next video.

Theory and Examples

[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.

h(x) = |x + 2| − |x − 3|, −∞ < x < ∞

Key Concepts

Absolute Value Functions

Graphing Piecewise Functions

Finding Extreme Values

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