Draw the graph of the function and discuss the continuity and differentiability at x = 1 for, `f(x) = {{:(3^(x)",","when",-1 le x le 1),(4-x",","when",1 lt x lt 4):}`

Answer

To solve the problem, we need to analyze the function given by:

\[ f(x) = \begin{cases} 3^x & \text{when } -1 \leq x \leq 1 \\ 4 - x & \text{when } 1 < x < 4 \end{cases} \]

### Step 1: Determine the Values of the Function at Key Points

1. **At \( x = -1 \)**:
\[ f(-1) = 3^{-1} = \frac{1}{3} \]

2. **At \( x = 0 \)**:
\[ f(0) = 3^0 = 1 \]

3. **At \( x = 1 \)**:
\[ f(1) = 3^1 = 3 \]

4. **At \( x = 2 \)** (for the second part of the function):
\[ f(2) = 4 - 2 = 2 \]

5. **At \( x = 3 \)**:
\[ f(3) = 4 - 3 = 1 \]

6. **At \( x = 4 \)**:
\[ f(4) = 4 - 4 = 0 \]

### Step 2: Plot the Graph of the Function

- For \( -1 \leq x \leq 1 \), the graph is represented by \( f(x) = 3^x \).
- For \( 1 < x < 4 \), the graph is represented by \( f(x) = 4 - x \).

### Step 3: Check Continuity at \( x = 1 \)

To check for continuity at \( x = 1 \), we need to verify if:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) \]

1. **Left-hand limit**:
\[ \lim_{x \to 1^-} f(x) = f(1) = 3 \]

2. **Right-hand limit**:
\[ \lim_{x \to 1^+} f(x) = 4 - 1 = 3 \]

Since both limits equal \( f(1) \), we conclude:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) = 3 \]

Thus, **the function is continuous at \( x = 1 \)**.

### Step 4: Check Differentiability at \( x = 1 \)

To check for differentiability at \( x = 1 \), we need to verify if:

\[ f'(1^-) = f'(1^+) \]

1. **Left-hand derivative**:
\[ f'(x) = \frac{d}{dx}(3^x) = 3^x \ln(3) \]
Evaluating at \( x = 1 \):
\[ f'(1^-) = 3^1 \ln(3) = 3 \ln(3) \]

2. **Right-hand derivative**:
\[ f'(x) = \frac{d}{dx}(4 - x) = -1 \]
Evaluating at \( x = 1 \):
\[ f'(1^+) = -1 \]

Since \( f'(1^-) \neq f'(1^+) \) (i.e., \( 3 \ln(3) \neq -1 \)), we conclude that:

**The function is not differentiable at \( x = 1 \)**.

### Conclusion

- The function \( f(x) \) is **continuous** at \( x = 1**. - The function \( f(x) \) is **not differentiable** at \( x = 1**.