So in this question we are told we have a function F of X with the following properties. And we want to answer the following questions about f of X. So first I want to do part D. Because if I sketch part D. If I make a sketch of this function having all these properties, I'll be able to see what's happening with A B and C. So let's make our sketch part D. And then we'll answer A B and C. They say f of X equals zero, has solutions at X equals eight. X equals -4 and no others. So that means I have X intercepts At x equals eight and X equals negative four. And those are my only X intercepts They say the limit as X approaches infinity of F of X is three. And the limit as X approaches negative infinity of F of X is three. That means I have what is called a horizontal A sento At Y. Equals three. They are saying that as I go to the right forever and ever My wise get closer and closer to three and as I go to the left forever and ever my wise are also getting closer and closer 23. They say that the limit as X approaches three from the left of this function is 10. So if I came way up here, Here's The .3, 10. And we are saying that as X gets closer and closer to three But stays to the left of three, my wise get closer and closer to 10. I also know That as X approaches five from the right, my wise approach negative infinity. As X approaches five from the left, my wise approach positive infinity. That means I have a vertical ascent. Okay, At x equals five. And they are saying That as X approaches five from the right, my wise approach negative infinity, I'm gonna move this graph down just a little bit so I can extend upwards. So here is x equals five. And they are saying that as X gets closer and closer to five from the right, my wise go down towards negative infinity As X gets closer to five from the left, my wise approach positive infinity. And finally, My function is continuous for all X values other than five. And so there are no holes, no breaks, no jumps in my graph. And so I think the only way that I could have all of these characteristics is to have a function that roughly speaking looks like this. So that's part D. Now let's go back and answer A B and C in a We want the limit as X approaches three from the right of this function and we're defending our answer by referring to f of X. So we know f is continuous everywhere besides X equals five. So that means specifically the fx is continuous At x equals three if f is continuous at X equals three. The limit As X approaches three from the left of my function has to equal the limit as X approaches three from the right of my function, They said the limit as X approaches three from the left was 10 and so 10 is going to have to be equal to the limit As X approaches three from the right of my function, there's party. Now I need be I want to find f of three and to defend my answer by referring to fx. Well, again they said the fx Is continuous at x equals three. By definition for F to be continuous At x equals three means That the limit as X approaches three of FX Equals there for three. But the limit as X approaches three of this function again is 10. And so I'm getting 10 equals and for three in C I want the sign of F of zero. Is it positive or negative? Well, looking at my graph here, F of zero is positive F of zero is positive. Why do I know this? Well, I know that there is only a zero To the left of that vertical ascent toe to the left of x equals five at X equals what was that? X equals negative four. And I know also F is continuous On the interval from negative 4 to 5. So what this means is since the limit As X approaches five from the left of my function with positive infinity, My f of zero had to be zero. That's my final answer here. Hopefully this makes sense. Have a great rest of your day.