https://file.aiquickdraw.com/mathbot/file/1726871953669u2mp8qtj.jpg

Let's work through the problem you've provided in the image. The piecewise function is given as:

\[
g(x) =
\begin{cases} 
-3x^2 + 8x & \text{if } x < 2 \\
a & \text{if } x = 2 \\
7x + 9 & \text{if } x > 2
\end{cases}
\]

The question is asking about the continuity of \( g(x) \) at \( x = 2 \). To ensure continuity, we need the left-hand limit, the function value, and the right-hand limit at \( x = 2 \) to be equal. This means:

\[
\lim_{x \to 2^-} g(x) = g(2) = \lim_{x \to 2^+} g(x)
\]

### Part (a): Left-hand limit as \( x \to 2^- \)
The expression for \( g(x) \) when \( x < 2 \) is \( -3x^2 + 8x \). We calculate the left-hand limit:

\[
\lim_{x \to 2^-} g(x) = -3(2)^2 + 8(2) = -3(4) + 16 = -12 + 16 = 4
\]

### Part (b): Right-hand limit as \( x \to 2^+ \)
The expression for \( g(x) \) when \( x > 2 \) is \( 7x + 9 \). We calculate the right-hand limit:

\[
\lim_{x \to 2^+} g(x) = 7(2) + 9 = 14 + 9 = 23
\]

### Part (c): Value of \( g(2) \)
To ensure continuity, we need:

\[
g(2) = a
\]

### Condition for Continuity
For \( g(x) \) to be continuous at \( x = 2 \), the left-hand limit, right-hand limit, and the value of \( g(2) \) must all be equal. This gives us the condition:

\[
4 = a = 23
\]

Since this condition is impossible (because \( 4 \neq 23 \)), there is **no value of \( a \)** that makes the function continuous at \( x = 2 \).

Would you like more details or have any questions on this? Here are 5 related questions that could further your understanding:

1. What does it mean for a function to be continuous at a point?
2. How do we evaluate limits from the left and right at a given point?
3. Can a function be continuous even if its piecewise parts are defined differently?
4. What would happen if the left-hand and right-hand limits were equal but the value at the point differed?
5. How does the behavior of quadratic and linear functions influence continuity at a point?

**Tip**: Always check both one-sided limits when determining the continuity of a piecewise function.

Let g(x) be a piecewise function: g(x) = -3x^2 + 8x if x < 2, g(x) = a if x = 2, g(x) = 7x + 9 if x > 2. Determine the value of a for which g(x) is continuous at x = 2.

Learn how to determine the value of a piecewise function's parameter 'a' such that g(x) is continuous at x = 2. Involves calculating left-hand and right-hand limits and applying the continuity condition. Suitable for students in Grades 10-12.

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