https://file.aiquickdraw.com/mathbot/file/17264672184680dzu3800.jpg

The function \( f(x) \) given in the image is a piecewise function:

\[
f(x) =
\begin{cases}
\frac{2x^2 - 5x - 3}{x - 3} & \text{if } x \neq 3 \\
6 & \text{if } x = 3
\end{cases}
\]
Additionally, we are provided that \( a = 3 \).

Let's analyze this:

1. **Simplification of \( \frac{2x^2 - 5x - 3}{x - 3} \) for \( x \neq 3 \)**:
   - First, factor the numerator \( 2x^2 - 5x - 3 \).
   - Find the roots of the quadratic to check if the factor \( x - 3 \) can be canceled out.

Let's proceed with the factorization of \( 2x^2 - 5x - 3 \).The quadratic expression \( 2x^2 - 5x - 3 \) factors as:

\[
2x^2 - 5x - 3 = (x - 3)(2x + 1)
\]

Thus, for \( x \neq 3 \), the function simplifies as:

\[
\frac{2x^2 - 5x - 3}{x - 3} = \frac{(x - 3)(2x + 1)}{x - 3} = 2x + 1
\]

Therefore, the simplified piecewise function is:

\[
f(x) =
\begin{cases}
2x + 1 & \text{if } x \neq 3 \\
6 & \text{if } x = 3
\end{cases}
\]

Next, we check the continuity of the function at \( x = 3 \) by ensuring that the value from the simplified function matches \( f(3) = 6 \).

Let’s evaluate the limit of \( 2x + 1 \) as \( x \) approaches 3.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can directly evaluate the simplified function \( f(x) = 2x + 1 \) at \( x = 3 \):

\[
f(3) = 2(3) + 1 = 7
\]

Since \( f(3) = 7 \) from the simplified expression, and the given value at \( x = 3 \) is \( f(3) = 6 \), this implies the function is **not continuous** at \( x = 3 \).

Would you like further details on this, or have any questions? Here are a few related topics to explore:

1. What is the definition of continuity for a piecewise function?
2. How to factor polynomials in general?
3. Why does the function break at \( x = 3 \)?
4. How does one check differentiability at a

Given the piecewise function f(x) = (2x^2 - 5x - 3) / (x - 3) if x ≠ 3 and f(x) = 6 if x = 3. Determine the continuity of the function at x = 3.

Explore the continuity of a piecewise function involving a quadratic factorization. Learn step-by-step how to simplify, factor, and apply limits to determine whether a function is continuous at a point. Suitable for advanced high school or early college students.

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