Math Problem Statement
Sketch the graph f(x) =
ex if x < 1 x3 if x ≥ 1 , a = 1
Solution
To sketch the graph of the piecewise function:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Exponential Functions
Cubic Functions
Continuity
Limits
Formulas
f(x) = e^x for x < 1
f(x) = x^3 for x ≥ 1
Theorems
Continuity and Discontinuity of Piecewise Functions
Limit Theorems
Suitable Grade Level
Grades 10-12
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