In the xy-plane, the graph of a rational function f has a hole at x=2. Input values of f sufficiently close to 2 correspond to output values arbitrarily close to 6 . Which of the following could define f(x) ?

To solve this problem, let's break down the key information given:

1. The graph of the rational function \( f \) has a hole at \( x = 2 \). This means that the function is undefined at \( x = 2 \), but is defined near \( x = 2 \).
2. For input values close to \( x = 2 \), the output values of the function \( f(x) \) approach 6. This indicates that \( \lim_{x \to 2} f(x) = 6 \).

We are asked to determine which of the following rational functions could define \( f(x) \).

### Concept: 
A hole at \( x = 2 \) usually happens when the function has a factor of \( (x - 2) \) in both the numerator and denominator, which cancels out but leaves the function undefined at \( x = 2 \). Additionally, the remaining function should give a limit of 6 as \( x \) approaches 2.

### Step-by-Step Approach:

Let’s say the rational function has the form:
\[
f(x) = \frac{(x - 2) \cdot P(x)}{(x - 2) \cdot Q(x)}
\]
where \( P(x) \) and \( Q(x) \) are polynomials.

Since there’s a hole at \( x = 2 \), the factor \( (x - 2) \) cancels out, but the limit as \( x \to 2 \) should give \( f(x) \to 6 \). After cancellation, the remaining function should approach 6 when \( x = 2 \).

So after canceling the factor of \( (x - 2) \), the simplified function would be:
\[
f(x) = \frac{P(x)}{Q(x)}
\]
and when \( x = 2 \), this must evaluate to 6:
\[
\frac{P(2)}{Q(2)} = 6
\]

Now, we can look at the possible rational functions to see which satisfies this condition.

Please provide the list of possible options so we can identify the correct function.

In the xy-plane, the graph of a rational function f has a hole at x=2. Input values of f sufficiently close to 2 correspond to output values arbitrarily close to 6. Which of the following could define f(x)?

Explore how to determine the rational function that has a hole at x=2 and approaches a value of 6 near this point. This problem involves key concepts in rational functions, limits, and the factor-cancellation theorem. Suitable for high school students in Grades 10-12.

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