Q. 29 - Calculus

Question

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

$$\begin{gathered} f has a jump discontinuity at x = -1 and is left continuous at \\ x = -1, and f (-1) = 2. \end{gathered}$$

Step-by-Step Solution

Answer

The graph of the given function is:

Step 1. Given Information.

Given the function has a jump discontinuity at $x=-1$.

It is left continuous at $x=-1.$

And also $f(-1) = 2$

Step 2. Drawing graph from the given conditions.

Now firstly point a coordinate on the graph which is (-1,2).

And now as we know the graph is left continuous draw any line or curve which is continuous to the left of the point we located on the graph as given in the diagram:

As it is clear from this graph we have $point A\equiv (-1,2)$ and the graph is left continuous at $x=-1.$

Step 3. Completing all the conditions to complete the graph.

Now we have one more condition to be satisfied with us which is our function has a jump discontinuity at the point A (-1,2). So, draw any continuous curve in the right side of the graph from the point A satisfying it should not touch the point A (-1,2).  As seen in the below graph:

So, this is our final graph of the function given satisfying all the conditions.

Note: This is not a unique function for the set of conditions given in the question there may be infinite type of functions we can draw from this.