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The problem presented in the image asks for the value of a constant \( a \) such that the given piecewise function \( f(x) \) is continuous for all values of \( x \).

The function is defined as:

\[
f(x) = 
\begin{cases} 
\frac{2x - 6a + 8}{x - 2} & \text{if } x \neq 2 \\
7 & \text{if } x = 2
\end{cases}
\]

### Step 1: Find the limit of \( f(x) \) as \( x \) approaches 2.

For the function to be continuous at \( x = 2 \), the limit of \( f(x) \) as \( x \) approaches 2 must equal the value of the function at \( x = 2 \), which is 7.

We first need to calculate the limit:

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

### Step 2: Simplify the function

Notice that the expression \( \frac{2x - 6a + 8}{x - 2} \) is indeterminate at \( x = 2 \), so we need to simplify the numerator. 

The numerator can be rewritten as:

\[
2x - 6a + 8 = 2(x - 2) - 6a + 12
\]

Thus,

\[
\frac{2x - 6a + 8}{x - 2} = \frac{2(x - 2) + (12 - 6a)}{x - 2}
\]

\[
= 2 + \frac{12 - 6a}{x - 2}
\]

To avoid the function being undefined, the term \( \frac{12 - 6a}{x - 2} \) must be zero, implying that:

\[
12 - 6a = 0
\]

### Step 3: Solve for \( a \)

Solving for \( a \) gives:

\[
6a = 12 \implies a = 2
\]

### Step 4: Verify continuity

If \( a = 2 \), the expression simplifies to \( f(x) = 2 \) for \( x \neq 2 \). Thus, the limit as \( x \) approaches 2 is:

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

But for \( f(x) \) to be continuous at \( x = 2 \), this limit must equal the value of \( f(2) = 7 \). Hence, the correct value of \( a \) should satisfy the condition \( f(x) = 7 \) as \( x \) approaches 2.

So, we need to revisit the problem. We seek a configuration where the expression is made to match 7 and not just zero out.

This indicates the problem requires re-analysing after ensuring the cancellation is properly accounted. We can then continue to explore the specific conditions where the piecewise parts converge.

Find the value of the constant 'a' for the function f(x) so that it is continuous for all real numbers. The function is defined as: f(x) = (2x - 6a + 8)/(x - 2) if x ≠ 2, and f(x) = 7 if x = 2.

Learn how to find the value of the constant 'a' to ensure the continuity of a piecewise function. This problem involves key mathematical concepts such as limits and continuity, with detailed steps for simplifying rational expressions and applying limit theorems. Suitable for students in Grades 10-12.

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