{Use of Tech} Triple intersection Graph the functions f(x) = x...

Question

{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

Accepted Answer

Below there, today we're gonna 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. Use a graphing utility to sketch the graphs of the following functions and identify their intersection point. P X equals X2, Q X equals 2 of X, and R of X is equal to 2 multiplied by X, all to the power of X divided by 2. Awesome. So it appears for this particular problem we're asked to take these three separate functions and we're asked to graph them using some sort of graphing utility or graphing software to graph these 3 separate functions. So when we do that, we will get 3 separate curves, and from these 3 separate curves, we're trying to figure out where all three of these curves intersect. So the point of intersection, where do all of these 3 curves meet and intersect with each other. So those are our two final answers we're ultimately trying to solve for is firstly, we're trying to create a graph of these three separate functions to produce 3 separate curves. And once we have that graph, we need to use that graph to help us figure out what the intersection point of these 3 curves are. So once again, to reiterate, so we're taking these 3 separate functions PFX, Q of X, and R of X. And we're going to be graphing them in our graphing utility or graphing software, and then we're going to be using that graph that we create to help us determine our intersection point. So with that said, so you can use whatever graphing software makes the most sense to you. So you can also use your graphing calculator, and if you need an idea of a free online graphing software to use online, you can always use Desmos. And desmos is super easy to use and you would just plug in these 3 separate functions to create the three separate curves. So when you've picked a graphing software or used your calculator, whatever makes sense to you, the graph that you should get when you plot those 3 functions into your graphing software, it should give you the following graph. In order, going from PF X to Q of X to R of X, we have our parabola, which represents our PFX function. And then for Q of X, and once again to quickly go back for a second, so for our parabola for PF X, which is X2d, it's represented in purple, that is our purple curve, it's the parabola. And for our Q of X function, which is going to be represented by the red curve, that is for 2 to the power of X, and finally our blue curve, which is our last and final curve, which is represented in blue, is denoting our R of X function, and that is 2 multiplied by X to the power of X divided by 2. And as you can see, we have going in order again, we have our parabola. And then for our and the problem once again is for our PFX function. And then for Q of X, we have what appears to be like an exponential growth kind of based curve, because as you can see, it starts from the negative side of our coordinate plane and it slowly increases, increases, increases going up into the positive quadrant of our coordinate plane. So it's showing growth, so it starts out steady and then it just starts to grow, grow and grow exponentially. And then lastly, for our RFX function, which is represented in blue, we have a, it starts from 0.1, so 01, and it looks like it dips down just a little bit, and then it starts to bounce back up. So almost like half of a parabola, almost, kind of the shape, but it starts from 01, and it dips down just a hair, and then it curves upwards, or tends upwards. So, Visually looking at this graph, we could see the point of intersection, and once again this is just looking at the graph just visually, we could see that the point of intersection. is simply going to be at 2.4, which I indicated with a green dot because I, as you can see, this is where all three curves intersect with each other. So that means the point of intersection will be at 2.4, which, as you should recall, 2.4 means X, Y, so that's our coordinate point. So 2.4, so 24. And that is our point of intersection, and that is our 2nd and final answer. Hooray, we did it. So once again, to quickly recap, so by inspecting our final graph that we get from when we plug in our curve, so when we plug in our functions to produce the curves in our graphing software, we will find when we inspect our graph, we will find that the point of intersection of these three separate functions will be 2.4. And that's it. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

Key Concepts

Graphing Functions

Intersection Points

Numerical Methods

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