Question

Let g be a function defined for all real numbers except for -3. Which of the following statements about g is true?

A) g is continuous at -3.
B) g is not defined at -3.
C) g is differentiable at -3.
D) g is a constant function.

Answer 1

In order for a function to be continuous at a point, it must be defined at that point. However, in this case, the function g is not defined at -3. This means that g is not continuous at -3, thus the correct option is B.

Step-by-step explanation:

In order for a function to be continuous at a point, it must be defined at that point. However, in this case, the function g is not defined at -3. This means that g is not continuous at -3.

To understand why g is not defined at -3, we first need to look at the definition of a function. A function is a rule that assigns each input (or independent variable) to exactly one output (or dependent variable). In other words, for every x-value we plug into the function g, we should get one and only one y-value.

Now, let's consider the given function g. It is defined for all real numbers except for -3. This means that for every x-value except for -3, we can plug it into the function g and get a unique y-value. However, when we try to plug in -3 as the input, we run into a problem. Since -3 is not in the domain of g, we cannot get a corresponding y-value. Therefore, g is not defined at -3.

We can also look at this graphically to further understand why g is not defined at -3. The graph of a function is a visual representation of all the input-output pairs of that function. In this case, the graph of g will have a 'hole' or a 'jump' at x = -3. This hole represents the fact that there is no output for the input -3. The graph may be continuous everywhere else, but it is not continuous at x = -3.

Furthermore, in order for a function to be differentiable at a point, it must also be continuous at that point. Since g is not continuous at -3, it cannot be differentiable at -3. This eliminates option C) as a possible correct answer.

In conclusion, the correct answer is B) g is not defined at -3. This is because g is not continuous at -3, it is not differentiable at -3, and it cannot be a constant function. The key concept to remember is that a function must be defined at a point in order for it to be continuous or differentiable at that point.