Find AA given that the function, f(x)f(x), is continuous at 4.

f(x)={x2Ax−8x<4x≥4f(x)={x2x<4Ax−8x≥4


6


7


2


4


5



To find the value of \( A \) such that the function \( f(x) \) is continuous at \( x = 4 \), let's consider the given function:

\[
f(x) = 
\begin{cases} 
x^2 & \text{if } x < 4 \\
Ax - 8 & \text{if } x \geq 4 
\end{cases}
\]

For \( f(x) \) to be continuous at \( x = 4 \), the left-hand limit as \( x \) approaches 4 from the left must equal the right-hand limit as \( x \) approaches 4 from the right, and both must equal the function value at \( x = 4 \).

### Step 1: Compute the left-hand limit as \( x \to 4^- \)
For \( x < 4 \), \( f(x) = x^2 \).
\[
\lim_{x \to 4^-} f(x) = 4^2 = 16
\]

### Step 2: Compute the right-hand limit as \( x \to 4^+ \)
For \( x \geq 4 \), \( f(x) = Ax - 8 \).
\[
\lim_{x \to 4^+} f(x) = A(4) - 8 = 4A - 8
\]

### Step 3: Set the limits equal to ensure continuity
For continuity at \( x = 4 \), the limits from both sides must be equal:
\[
\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x)
\]
\[
16 = 4A - 8
\]

### Step 4: Solve for \( A \)
Now, solve the equation \( 16 = 4A - 8 \):
\[
16 + 8 = 4A
\]
\[
24 = 4A
\]
\[
A = \frac{24}{4} = 6
\]

### Conclusion
The value of \( A \) that makes the function \( f(x) \) continuous at \( x = 4 \) is \( \boxed{6} \).

Would you like a further explanation on this? Here are some related questions:

1. What are the conditions for a function to be continuous at a point?
2. How do you handle continuity at points where the function changes its form?
3. Can a function be continuous but not differentiable at a point?
4. How do you find the limit of more complex piecewise functions?
5. What is the significance of ensuring continuity in a function?

**Tip:** Always equate the left-hand limit and right-hand limit at the point of interest to ensure continuity in piecewise functions.

Learn how to find the value of A that ensures the function f(x) = { x^2 for x < 4, Ax - 8 for x ≥ 4 } is continuous at x = 4. Follow a step-by-step solution involving limits and continuity conditions. Suitable for advanced high school students and anyone interested in calculus concepts.

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