Given 𝑓(𝑥)={2𝑥^2+𝑎𝑥−6,𝑥<5 62−𝑎𝑥 ,𝑥≥5 determine the value of “a” that makes the function continuous

To make the function \( f(x) \) continuous at \( x = 5 \), the left-hand limit as \( x \to 5^- \) (approaching from the left) must equal the right-hand limit as \( x \to 5^+ \) (approaching from the right). Additionally, the function's value at \( x = 5 \) must match both limits.

Given:

\[
f(x) = 
\begin{cases}
2x^2 + ax - 6 & \text{for } x < 5 \\
62 - ax & \text{for } x \geq 5
\end{cases}
\]

### Step 1: Calculate the left-hand limit as \( x \to 5^- \)
From the first piece of the function, \( f(x) = 2x^2 + ax - 6 \) for \( x < 5 \).

The left-hand limit as \( x \to 5^- \) is:
\[
\lim_{x \to 5^-} f(x) = 2(5)^2 + a(5) - 6 = 50 + 5a - 6 = 44 + 5a
\]

### Step 2: Calculate the right-hand limit as \( x \to 5^+ \)
From the second piece of the function, \( f(x) = 62 - ax \) for \( x \geq 5 \).

The right-hand limit as \( x \to 5^+ \) is:
\[
\lim_{x \to 5^+} f(x) = 62 - a(5) = 62 - 5a
\]

### Step 3: Set the left-hand and right-hand limits equal to each other
For the function to be continuous, the limits must be equal at \( x = 5 \). So:
\[
44 + 5a = 62 - 5a
\]

### Step 4: Solve for \( a \)
Add \( 5a \) to both sides:
\[
44 + 10a = 62
\]
Subtract 44 from both sides:
\[
10a = 18
\]
Divide by 10:
\[
a = \frac{18}{10} = 1.8
\]

Thus, the value of \( a \) that makes the function continuous is \( a = 1.8 \).

Would you like further details on the solution or have any questions?

Here are 5 related questions to explore:

1. How do we ensure continuity at any other point in a piecewise function?
2. What is the geometric interpretation of continuity at a point?
3. How can discontinuity occur in a function, and what types are there?
4. What does it mean for a function to be differentiable at a point, and how does it relate to continuity?
5. Can a function be continuous at all points but not differentiable at some points?

**Tip:** Continuity at a point requires that the function's value and both one-sided limits agree at that point.

Given 𝑓(𝑥)={2𝑥^2+𝑎𝑥−6,𝑥<5 62−𝑎𝑥 ,𝑥≥5 determine the value of 'a' that makes the function continuous

Learn how to determine the value of 'a' that makes a piecewise function continuous at x = 5. This problem explores key concepts like limits, continuity, and piecewise functions, and includes a detailed solution. Ideal for students in Grades 11-12.

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