Question

Sketch a graph of a function \(f\) that satisfies the given values. (There are many correct answers.) $$ \begin{aligned} &f(-2)=0\\\ &f(2)=0\\\ &\lim _{x \rightarrow-2} f(x)=0\\\ &\lim _{x \rightarrow 2} f(x) \text { does not exist. } \end{aligned} $$

Short answer

The graph of the function should touch the x-axis at x=-2, show point discontinuity at x=2 (with a open circle at the point (2,0)), and should not cross or touch the x-axis anywhere between x=-2 and x=2.

Step-by-step solution

Graphing given points

Start by plotting the points for which we know the function's value: \((-2, 0)\) and \((2, 0)\).

Understanding the limits

The limit of the function as \(x\) approaches \(-2\) exists, and is equal to the function's value at that point, suggesting continuity at this point. So, the graph should approach and touch the x-axis at \(x= -2\). However, the limit as \(x\) approaches \(2\) does not exist, which indicates some kind of discontinuity at \(x = 2\). The function could either jump or have an asymptote at that point. For this exercise, let's assume that there is a point discontinuity at \(x = 2\).

Sketching the function

On the interval \(-\infty < x \leq -2\), draw the function approaching the point \((-2, 0)\) from the left. Then, from \(x = -2\) to \(x = 2\), sketch the function such that it does not touch or cross the x-axis until \(x = 2\). At \(x = 2\), show a point discontinuity by leaving an open circle at the point \((2, 0)\). Lastly, sketch the function for \(2 < x < \infty\) as per your understanding.