What function value must be assigned for f(2)f(2) so that the following function is a continuous function for all real numbers?

f(x)=x2−6x+8x−2f(x)=x2−6x+8x−2

If x≠2

To make the function \( f(x) \) continuous for all real numbers, including \( x = 2 \), we need to determine the appropriate value of \( f(2) \).

### 1. Simplify the Function
First, simplify the expression for \( f(x) \) when \( x \neq 2 \):

\[
f(x) = \frac{x^2 - 6x + 8}{x - 2}
\]

Factor the quadratic expression in the numerator:

\[
x^2 - 6x + 8 = (x - 2)(x - 4)
\]

So the function simplifies to:

\[
f(x) = \frac{(x - 2)(x - 4)}{x - 2}
\]

For \( x \neq 2 \), the factor \( (x - 2) \) cancels out, and we are left with:

\[
f(x) = x - 4 \quad \text{for } x \neq 2
\]

### 2. Determine the Limit as \( x \) Approaches 2
To ensure continuity at \( x = 2 \), we need to find the limit of \( f(x) \) as \( x \) approaches 2:

\[
\lim_{x \to 2} f(x) = \lim_{x \to 2} (x - 4) = 2 - 4 = -2
\]

### 3. Assign the Value of \( f(2) \)
For \( f(x) \) to be continuous at \( x = 2 \), the value \( f(2) \) must be equal to the limit found above:

\[
f(2) = -2
\]

### Conclusion
To make \( f(x) \) continuous for all real numbers, we must assign:

\[
f(2) = -2
\]

This ensures that the function is continuous at \( x = 2 \) and everywhere else.

Would you like further details or have any questions? Here are some related questions:

1. What is the definition of a continuous function?
2. How do you determine if a piecewise function is continuous at a specific point?
3. Can a function have a removable discontinuity? How does that relate to this problem?
4. What happens if we choose a different value for \( f(2) \)? 
5. How do you find limits involving piecewise functions?

**Tip:** When a function has a potential discontinuity due to division by zero, try simplifying the function and checking the limit at the problematic point.

Discover how to ensure continuity of the function f(x) = (x^2 - 6x + 8) / (x - 2) by determining the value of f(2). Learn about limits, continuity in functions, and solving piecewise functions to achieve a continuous function. Suitable for advanced high school students.

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