Does f(2) Exist? Evaluating Statements to Determine Existence of f(2)

Which of the following statements, if true, can be used to conclude that f(2) exists?

In order to determine if f(2) exists, we need to evaluate the given statements and see if any of them provide enough evidence to conclude the existence of f(2)

In order to determine if f(2) exists, we need to evaluate the given statements and see if any of them provide enough evidence to conclude the existence of f(2). Let’s analyze each statement individually:

1. f(x) is continuous at x = 2.

This statement implies that the function f(x) is continuous at x = 2. If a function is continuous at a specific point, it means that the function exists and has a well-defined value at that point. Therefore, this statement can be used as evidence to conclude that f(2) exists.

2. f(x) is differentiable at x = 2.

This statement implies that the function f(x) is differentiable at x = 2. Differentiability is a stronger condition than continuity, as it not only requires the function to exist but also requires the existence of the derivative at that point. Therefore, if a function is differentiable at x = 2, it necessarily exists at that point. Hence, this statement can be used as evidence to conclude that f(2) exists.

3. f(2) is the maximum value of the function.

This statement does not provide enough evidence on its own to conclude that f(2) exists. Even if f(2) is the maximum value of the function, it does not guarantee the existence of the function at that point. The maximum value could occur at a limit that exists as x approaches 2, but not necessarily at x = 2 itself.

4. f(x) has a vertical asymptote at x = 2.

Having a vertical asymptote at x = 2 suggests that the function approaches infinity or negative infinity as x approaches 2. However, the existence or non-existence of f(2) cannot be directly inferred from this statement. A function may have a vertical asymptote at a particular value, yet still exist or not exist at that point.

In conclusion, the statements that can be used to conclude that f(2) exists are:

1. f(x) is continuous at x = 2.
2. f(x) is differentiable at x = 2.

These conditions ensure not only the existence but also the well-defined nature of the function at x = 2, allowing us to conclude that f(2) exists.

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