a. Graph the function


ƒ(x) = { x², -1 ≤ x < 0
{ -x², 0 ≤ x ≤ 1.


b. Is ƒ continuous at x = 0?
c. Is ƒ differentiable at x = 0?


Give reasons for your answers.

Verified step by step guidance

1. Step 1: To graph the function ƒ(x), first note that it is a piecewise function with two parts: ƒ(x) = x² for -1 ≤ x < 0 and ƒ(x) = -x² for 0 ≤ x ≤ 1. Plot the graph for each piece separately within their respective intervals.
2. Step 2: For the interval -1 ≤ x < 0, plot the graph of x². This is a parabola opening upwards, but only consider the portion from x = -1 to just before x = 0.
3. Step 3: For the interval 0 ≤ x ≤ 1, plot the graph of -x². This is a parabola opening downwards, starting at x = 0 and ending at x = 1.
4. Step 4: To determine if ƒ is continuous at x = 0, check the left-hand limit (as x approaches 0 from the left) and the right-hand limit (as x approaches 0 from the right). Both limits must equal ƒ(0) for the function to be continuous at x = 0.
5. Step 5: To determine if ƒ is differentiable at x = 0, check if the derivative from the left-hand side (using x²) and the derivative from the right-hand side (using -x²) are equal at x = 0. If they are not equal, the function is not differentiable at x = 0.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function ƒ(x) has two distinct parts: x² for -1 ≤ x < 0 and -x² for 0 ≤ x ≤ 1. Understanding how to evaluate and graph these segments is crucial for analyzing the function's behavior at specific points, particularly at the transition point x = 0.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For ƒ to be continuous at x = 0, we need to check if the left-hand limit (as x approaches 0 from the left) equals the right-hand limit (as x approaches 0 from the right) and if both equal ƒ(0). This concept is essential for determining the function's smoothness at x = 0.

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which requires the function to be continuous there. To check differentiability at x = 0 for ƒ, we must evaluate the left-hand and right-hand derivatives. If these derivatives are not equal, the function is not differentiable at that point, indicating a potential corner or cusp in the graph.