Sketch the graph f(x) =

ex	    	if x < 1
x3		if x ≥ 1
,         a = 1

To sketch the graph of the piecewise function:

\[
f(x) = 
\begin{cases} 
e^x & \text{if } x < 1 \\
x^3 & \text{if } x \geq 1
\end{cases}
\]

### Step-by-Step Analysis:

1. **For \( x < 1 \):** The function is \( f(x) = e^x \).  
   - This is the standard exponential function, which increases rapidly as \( x \) increases.
   - As \( x \to -\infty \), \( e^x \to 0 \).
   - At \( x = 1 \), the function approaches \( e^1 = e \approx 2.718 \).

2. **For \( x \geq 1 \):** The function is \( f(x) = x^3 \).
   - This is a cubic function, which is increasing for all \( x \).
   - At \( x = 1 \), \( f(1) = 1^3 = 1 \).
   - As \( x \to \infty \), the cubic function grows more slowly compared to the exponential function.

### Continuity at \( x = 1 \):
- To check if the function is continuous at \( x = 1 \), we compare the values of the two pieces at this point.
   - From the exponential side: \( \lim_{x \to 1^-} f(x) = e^1 = e \).
   - From the cubic side: \( f(1) = 1^3 = 1 \).
   
Since these values are not equal, there is a **discontinuity** at \( x = 1 \).

### Graph Description:
- For \( x < 1 \), the curve follows the shape of \( e^x \), rising exponentially, and approaches the value \( e \) as \( x \to 1 \).
- At \( x = 1 \), there is a **jump discontinuity**: the function jumps from \( e \) down to \( 1 \).
- For \( x \geq 1 \), the function follows the shape of \( x^3 \), which is a smooth, increasing curve starting from \( f(1) = 1 \).

Would you like a visual graph representation or further clarification?

### 5 Related Questions:
1. What are the key properties of the exponential function?
2. How do you determine if a function is continuous at a given point?
3. What is the behavior of cubic functions as \( x \to \infty \)?
4. Can a function be differentiable if it is not continuous at a point?
5. How can you find limits from the left and right of a given point?

**Tip:** Always check both limits from the left and the right to verify the continuity of piecewise functions.

Sketch the graph f(x) = e^x if x < 1 and x^3 if x ≥ 1, a = 1.

Learn how to graph the piecewise function f(x) with e^x for x < 1 and x^3 for x ≥ 1. Explore key mathematical concepts such as exponential and cubic functions, continuity, and limits. This solution includes step-by-step analysis suitable for Grades 10-12 students.

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