Draw the graph of any function, f(x), satisfying the following conditions.

Question

1. Draw the graph of any function, f(x), satisfying the following conditions.

lim_x→-2 f(x) = 5

f(-2) = 1

lim_x→1- f(x) = 3

lim_x→1+ f(x) = -∞

I am aware you can't exactly "draw" a function on a website formatted as such, but would someone be able to explain a type of function that would satisfy these requirements?

Answer 1

Hi, Noah!

Like you said, I can't quite draw a function here, but I will attempt to describe what a function f(x) needs to look like given the specific conditions you are given.

1. To meet the first condition, you need the function to approach y = 5 from both the left and the right at the x-value x = -2.
2. However, to meet the second condition, you cannot have the function actually equal y = 5 at x = -2, because we are given f(-2) = 1. Therefore, you need to have an open circle at the point (-2, 5) and a closed circle at the point (-2, 1). This indicates that the point (-2, 5) is not actually included in the function (even though it must approach this point from both sides), and that the point (-2, 1) is included.
3. The - symbol to the upper right of the x = 1 in the limit indicates that the function must approach y = 3 from the left of the x-value x = 1. You then have the choice of including an open or closed circle at y = 3, though a closed circle would likely be easier here so that the function is defined at y = 3.
4. The + symbol to the upper right of the x = 1 in the limit indicates that the function must approach positive infinity from the right of the x-value x = 1.

Outside of these specific conditions, the function can look like whatever you like!

I hope this explanation was helpful, and please don't hesitate to reach out if you have any further questions!