https://file.aiquickdraw.com/mathbot/file/1729839014876j2h6fri0.jpg

The problem in the image asks to determine the value of \( a \) such that the function \( f(x) \) is continuous. The function is defined as:

\[
f(x) = \begin{cases} 
3x + 4, & x \leq a \\
x + 3, & x > a
\end{cases}
\]

To ensure the function is continuous at \( x = a \), the left-hand limit (as \( x \to a^- \)) must equal the right-hand limit (as \( x \to a^+ \)), and both must equal \( f(a) \).

### Step 1: Left-hand limit (\( x \leq a \))
For \( x \leq a \), the function is \( f(x) = 3x + 4 \). So the limit as \( x \to a^- \) is:
\[
\lim_{x \to a^-} f(x) = 3a + 4
\]

### Step 2: Right-hand limit (\( x > a \))
For \( x > a \), the function is \( f(x) = x + 3 \). So the limit as \( x \to a^+ \) is:
\[
\lim_{x \to a^+} f(x) = a + 3
\]

### Step 3: Continuity condition
For \( f(x) \) to be continuous at \( x = a \), the two limits must be equal:
\[
3a + 4 = a + 3
\]

### Step 4: Solve for \( a \)
Now, solve this equation for \( a \):
\[
3a + 4 = a + 3
\]
\[
3a - a = 3 - 4
\]
\[
2a = -1
\]
\[
a = -\frac{1}{2}
\]

Thus, the value of \( a \) that makes the function continuous is \( a = -\frac{1}{2} \).

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:
1. What does it mean for a function to be continuous at a point?
2. How do you check continuity for piecewise functions at a point?
3. What happens if the function isn't continuous at the point where the rule changes?
4. Can you explain the difference between one-sided limits and two-sided limits?
5. What are some practical applications of continuous functions in real-world scenarios?

**Tip**: Continuity of a function ensures there are no sudden jumps or breaks, which is important in modeling real-world processes!

Tentukan a yang memenuhi agar fungsi di bawah ini kontinu: f(x) = {3x + 4, x ≤ a; x + 3, x > a}

Solve the continuity problem for the piecewise function f(x) = {3x + 4, x ≤ a; x + 3, x > a}. This solution explains how to use limits and continuity conditions to find the value of a that makes the function continuous.

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