Question

use the graph of y = f(x) shown above to answer the following: if you are having a hard time seeing the picture clearly, click on the picture. it will expand to a larger picture on its own page so that you can inspect it more clearly. (a) f(x) continuous at x = -1.

Explanation:

To determine if \( f(x) \) is continuous at \( x = -1 \), we check the three conditions for continuity:

Step 1: Check if \( f(-1) \) is defined

From the graph, at \( x = -1 \), there is a blue dot (closed circle), so \( f(-1) \) has a defined value.

Step 2: Check the limit as \( x \to -1 \)

As \( x \) approaches \( -1 \) from the left (left - hand limit) and from the right (right - hand limit), we observe the behavior of the graph. The left - hand limit (as \( x \) approaches \( -1 \) from values less than \( -1 \)) and the right - hand limit (as \( x \) approaches \( -1 \) from values greater than \( -1 \)) should be equal for the limit to exist. From the graph, the left - hand limit (approaching \( x=-1 \) from the left) and the right - hand limit (approaching \( x = - 1\) from the right) appear to be equal (the graph approaches the same \( y \) - value from both sides).

Step 3: Check if \( \lim_{x\to - 1}f(x)=f(-1) \)

Since \( f(-1) \) is defined, and the left - hand limit and right - hand limit as \( x\to - 1 \) are equal (so the overall limit \( \lim_{x\to - 1}f(x) \) exists) and equal to \( f(-1) \), the function satisfies all three conditions for continuity at \( x=-1 \).

Answer:

\( f(x) \) is continuous at \( x = - 1 \) (the blank should be filled with "is" if we assume the dropdown is between "is" and "is not", based on the graph analysis).