[Technology Exercise]

a. Gra...

Question

[Technology Exercise]

a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which

x/2 > 1 + 4/x

b. Confirm your findings in part (a) algebraically.

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the X values for which X is greater than 1 + 20 divided by X by graphing the functions F of X equals X and G of X is equal to 1 + 20 divided by x together. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answers we're trying to find the X values for which X is greater than 1 + 20 divided by X. That's our first answer. And our second answer is we're trying to create a graph of the functions F of X and G of X, and we're trying to plot them on the same coordinate plane on the same graph together. And we're also told a hint which we can find the X values by graphing our functions of F of X and G of X. Awesome. So now that we know what we're ultimately trying to solve for, our first step is we need to graph our function of F of X is equal to X, and as we should be able to recognize and recall, this will be a straight line that passes through our origin point, which is 0.0. And it will have a slope of 1, so meaning it will pass through the following coordinate points. So it'll be passing through our curve, or rather this linear curve, it's going to pass through the coordinate points, 0.0, the origin point. 11. And lastly, it will also pass through the point -1.com. -1, so -1.com -1. So it'll pass through the following coordinate points for this particular curve. So, with that said, Our second step now is we need to graph our function for G of X. So let us note that our graph of G of X, which G of X is going to be equal to 1 + 20 divided by X. So, as we should be able to recall and recognize, this particular function will represent a hyperbola. With a vertical asymptote at X equals 0. And once again, this vertical asymptote at X equals 0 will be where the function is undefined, and it will also have a horizontal asymptote of Y equals 1. And we also need to note that for X is greater than 0, the function will decrease as X increases, and for when X is less than 0, the function will also decrease as X becomes more negative. So with that said, let us also note that when X equals 1, that will mean that G of 1, meaning we're plugging in a value of 1 in for X, will be equal to 1 + 20 divided by 1, which will be equal to 21, and for when X equals 2, when X equals 4 when X equals. 5 and X equals -1, when X equals -2. And when X equals -4, and when X equals -5. So in an effort to save time, I'm gonna skip a few steps. So, once again, we're plugging in different values for X. So instead of rewriting the entire expression out every single time, I'm just going to give you what the value of G of blank, what the new value of G of X will be when you plug in a when you plug in these specific values of X. So when X is equal to 2, that will mean that the function for G of X is going to be equal to 11, and when X is equal to 4, it's going to be equal to 6. When X is equal to 5, it's going to be equal to 5. When X is equal to -1, it's going to be equal to -19. When X is equal to -2, it's going to be equal to -9. When X is equal to -4. GFX is going to be equal to -4, and when X is equal to -5, GFX is gonna be equal to -3. Fantastic. So, these will help us plot our different points on our graph to help us graph our function of GFX, which let us note that when we go to graph our curves, we're going to graph our curve for F of X in red, and we're going to curve, we're going to draw the curve for G of X to be in green. So our green curve is going to be for GFX and our red curve is going to be for F of X. Fantastic. So using our information that we compiled together, let's create a graph of our two different curves for F of X and G of X using your graphing software of choice. So I've gone ahead and plugged those two equations into my graphing software, and I have produced the following graph. As you can see, like I mentioned, our curve for F of X, which is our linear graph, is represented in red. With our three different coordinate points that we listed for -1-1, 0.0, and 1.1. We also have our curve for GFX represented in green, and we have our different coordinate points which we solved together. So we have 2.11, 4.6, 5.5. We also have -5.3, 4.4, -2.9, and -119. So now, our next step, which is our 3rd step, is we need to find our intersection points. So let's put a box around our graph so we know that we've already solved for one of our answers. So we've already solved for one answer we're trying to solve for. So now we're trying to solve for the values of X, for which X is greater than 1 + 20 divided by X. So now in order to do that, we need to find our intersection points. So in order to do that, we need to find where F of X is equal to G of X. We need to find when F of X is equal to G of X. So we need to solve where X is equal to 1 + 20 divided by X. So we need to multiply through by X and we also need to assume that X cannot equal 0. So with that in mind, We need to note that X2 is going to be equal to X + 20. Fantastic. So now we need to rearrange this expression to set it equal to 0, so we need to say that X2 minus X minus 20 is equal to 0. So now we need to solve the quadratic function by factoring. So when we do that, when we factor it out, we will find that X +4, which I'm going to skip a few steps. So using a little bit of algebra to factor, we'll find that X + 4 multiplied by X minus 5 will be equal to 0. So that will mean, without a doubt, that the graphs will intersect at, so let's make a note of this. So our graphs will intersect at X equals -4 and X equals 5. So now our fourth step is we now need to analyze our inequality. So our inequality that we're trying to analyze is where X is greater than 1 + 20 divided by X. So from our graph we can see that for when X is less than -4, that will mean that F of X is below G of X. So that will mean that X is less than 1 + 20 divided by X. So, for when -4 is less than X and X is less than 0. That will mean that F of X is above G of X, so that will mean that X is greater than 1 + 20 divided by X. And moving right along here. So when 0 is less than X and X is less than 5, that will mean that F of X is below G of X. So that will mean that X is less than 1 + 20 divided by X. And finally, when X is greater than 5, that will mean that F of X is above G of X. So that will mean that X is greater than 1 + 20 divided by X. So, therefore, without a doubt, our second and final answer that we're ultimately trying to solve for will be that, and let's write our final answer. In green. So that will mean, without a doubt that our final answer will be parentheses -40 5 infinity. Fantastic. And that's it. That is our final pair of answers. So we're asked to figure out to find the values of, we need to find X values for which this inequality is proven. So when X is greater than 1 + 20 divided by X, and that will be when parentheses -4.0 5 infinity. is our only correct answer for that particular range of values. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

[Technology Exercise]

a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which

x/2 > 1 + 4/x

b. Confirm your findings in part (a) algebraically.

Key Concepts

Graphing Functions

Inequalities

Algebraic Manipulation

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