Functions and Graphs

Find th...

Question

Functions and Graphs

Find the natural domain and graph the functions in Exercises 15–20.

g(x) = √−x

Accepted Answer

Hello there. Today we're going to solve for 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. Identify the domain and sketch the graph of the function. PFX is equal to the square root of -3 X. Fantastic. So it appears for this particular problem, we're asked to figure out what the domain of this function of PFX is, and we're also asked to graph this function of PFX. So with that said, First off, in order to determine the domain of our function p of X is equal to the square root of -3 X, we need to ensure that the expression inside of our square root symbol is a non-negative value, since the square root of a negative number will produce a not real number. So in order to find the domain, we need to note that the expression inside of the square root is -3X. So for it to be valid for real numbers, that will have to mean that -3 X has to be greater than or equal to 0. So with that said, now that we know that -3 -3 X has to be greater than or equal to 0, we now need to solve for X. So we need to divide both sides by -3, and we also need to remember to reverse the inequality sign. So when we do that, we will find that X is less than or equal to 0. So therefore, without a doubt, the domain of PF X is X is less than or equal to 0, which we can write this in inable notation as parentheses negative. Infinity, 0 square bracket. Fantastic. So to quickly recap here, so our domain states that the domain of PF X has to be X is less than or equal to 0, and we can write this in interval interval notation as parenthesis negative infinity square bracket. So a parenthesis indicates an open interval, whereas a square bracket indicates a closed interval. So with that said, We now need to figure out some pieces of information before we could start sketching our graph. So we need to figure out the intercepts, because the intercepts will help us plot our graph, or rather help us plot our curve. So with that said, we need to figure out the intercepts, so we need to note that when X is equal to 0, You will get P 0 is going to be equal to the square root of -3 multiplied by 0, which will mean that P 0 is going to be simply equal to 0. So therefore, the function will pass through the point 0.0 in parentheses. Now we need to determine the behavior of negative X. So when X is equal to -3. We need to note that P-3, so we're plugging in a value of -3 in for X, is going to be equal to the square root of -3 multiplied by -3, which will give us an answer of 9. And for when X is equal to -12, so let's say that X is equal to -12, we will find that P-12, so I don't have to rewrite the the original function every single time plugging the value in. I'm just gonna give you the answer. So when X is equal to -12, P of -12 is gonna be simply equal to 6, and when X is equal to negative 27. That will mean that P of -27 is gonna be simply equal to 9. Fantastic. So, with that said, we can now determine the end behavior. So we need to note that as X becomes more negative, -3 X will become larger, and that will mean that the square root function increases. So with that said, we can now note that the plot the points that we're gonna need to plot are gonna be at the origin point 0.0. We also note the point 3.3 and also the -12.6, and finally our coordinate point -27.9, because we found four different data points. So now we need to plot these 4 different points on our graph and then connect them with a smooth curve that passes through these points in order to help us graph the function of PFX equals the square root of -3 X, and we need to note that our curve that we will create will resemble a square root function. But it will be reflected across the y axis due to the negative coefficient located inside of the square root. So with that said, when we create a graph of our function, we should create it should look something like the following. So when we finally plot our points, our curve, which is represented in blue, I went ahead and plotted our function of PFX into some graphing software, which you can do the same to double check your answer. Now as you can see, we have our curve represented in blue, which, as we stated is a. It will resemble a square root function curve, but it's reflected across the Y axis, due to the negative symbol within or rather inside of our square root symbol. And as you can see, we have our one corner point which starts at the origin, which is 00, and it starts to curve upwards where it strikes our second point, which is -3.3. This increases up to -12.6, and then increases again to -27.9, and then stretches off the graph. Fantastic. So it looks like an arm that's curving to the left, and that's it. We've officially solved for this problem. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Functions and Graphs

Find the natural domain and graph the functions in Exercises 15–20.

g(x) = √−x

Key Concepts

Natural Domain

Square Root Function

Graphing Functions

Watch next