Piecewise linear functions Graph the...

Question

Piecewise linear functions Graph the following functions.

$f (x) = {\begin{matrix} 3 x - 1 \frac{}{}, if x \leq 0 \\ - 2 x - 1, if x > 0 \end{matrix}$

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. Sketch the graph of the following piece-wise function. U of X is equal to curly bracket of 2 x minus 5 if X is less than or equal to 0 and negative X minus 5 if X is greater than 0. OK. So it appears for this particular problem, we're asked to take our piece piecewise function for UF X and we're asked to draw a sketch for it. So in order to do that, our first step is we need to focus on our two separate functions that are within this piece-wise function, and we need to focus on them individually in order to help us get an overall idea of what the sketch. All together when it's put both together. So when we put both pieces, I like the name implies piece wise function when we take both of these functions separately and we put them together in the same graph. So right now let's just focus on each function separately. So let's start with 2 X minus 5 focusing on the condition that if X is less than or equal to 0. So with that in mind, let us consider the first condition where X is equal to 0 because X is less than or equal to 0. So when X is equal to 0, For that condition, we could say that U of X is equal to 2 multiplied by 0 because we plug in 0. 4 X. So 2 multiplied by 0 is 0, minus 5, which means that our final answer is gonna be -5, and similarly for you of X. Which we're gonna be focusing now once again, we're dealing with U of X is equal to 2 X minus 5. So we can make a little note of here up top here that we're dealing with U of X is equal to 2X minus 5. So similarly, if we focus on our second condition now when X is less than 0, so let's say that X is equal to -2 now. So similarly, like we did above for X equals 0, we'll take U of X is equal to 2 multiplied by -2, which will give us -4 minus 5, so -4 -5 will give us -9 as an answer. Fantastic. So now we can focus on our different coordinate points that our line needs to pass through based on this information. So now, considering what we know, we need to draw a line for our sketch that passes through the coordinate.0.5. And once again, this is for our X Y layout. This is our coordinate point. So we need to make sure we draw a line that passes through 0.5, and it also needs to pass through -29. And we also need to note, and let's call this capital R to represent right, so we need to note that the right End point, which I'll call EP endpoint. So the right endpoint. Of our line should be at 0.5 for our coordinate point. So once again the right endpoint of our line for our first function will be, which our first function, so this is our first curve, will be at 0.5. Fantastic. Now we could start focusing on our second function, which is for negative X minus 5. And let us know that X is going to be greater than 0. So let us say for our first since we're gonna be working with X for two different values like we did for our first function. So let's say for our first condition X is equal to 1. So let us know that we're now dealing with U of X is equal to negative X minus 5 now. So, when we say that U of X is equal to Negative 1 means we're plugging in 1 for X so negative and then in parentheses we put 1 in for X minus 5, so -1, minus 5 is equal to -6. And once again you of X, and let's say for our second value. And before we write you of X, let's take a moment here to consider a different value for X. So let's say that X is equal to 3 in this case, so we could write that U of X is equal to negative plugin or value for 3 for X, and it will be -3 minus 5, so -3 minus 5 is going to be equal to -8. So now using the information we gathered, we now need to recall a note that we need to draw a line that passes through the coordinate point 1.6, and it also needs to pass through a coordinate point of 3.8. And we also need to recall a note for this particular case that the line for this particular curve should be drawn only to the right side of the point. So this is, let's make a note of this for RS right side, and this is going to be only drawn to the right side of the coordinate point 0.5. So with that in mind, we now can plot our graphs so we could use graphing software to create a sketch of our graph, or we could do it by hand. I've gone ahead and plugged in our two different functions into some graphing software to produce the following graph. And as we could see, it follows all the conditions that we found together for our coordinate points. So as we could see, we have our 0.5, which appears to be the vertex of our two different curves here. And it also, as we could see following our different coordinate points, the line passes through -2, -9. Also, once again, the right end point is at 0.5. So that means that it stops at the right side. So that means that our second function, which is for the negative X minus 5, starts from the. Starts on the left and then goes towards the right. And then we have it starts on the left and goes down, so it goes up one side to make like a peak on the mountain and then it goes down the other. So we have two curves that are represented in red that are for the two separate curves for our piece-wise function that make up the one graph for U of X. So as you can see, we have our 0.1 -6 and then 38. And it passes through both of those points. 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.

Piecewise linear functions Graph the following functions.
$f (x) = {\begin{matrix} 3 x - 1 \frac{}{}, if x \leq 0 \\ - 2 x - 1, if x > 0 \end{matrix}$

Key Concepts

Piecewise Functions

Graphing Linear Functions

Continuity and Discontinuity

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