Math Problem Statement
Find the x-value at which f is discontinuous and determine whether f is continuous from the right, from the left, or neither.
Solution
Let's go through the problem step by step.
The function is defined in a piecewise manner:
\begin{cases} 2 + x^2 & \text{if } x \leq 0, \\ 4 - x & \text{if } 0 < x \leq 4, \\ (x - 4)^2 & \text{if } x > 4. \end{cases}$$ ### Part (a): Find the x-value at which $$f$$ is discontinuous. To find where the function might be discontinuous, we need to check the points where the pieces of the function switch, i.e., at $$x = 0$$ and $$x = 4$$. #### Checking $$x = 0$$: - **Left-hand limit** as $$x \to 0^-$$: $$\lim_{x \to 0^-} f(x) = 2 + (0)^2 = 2.$$ - **Right-hand limit** as $$x \to 0^+$$: $$\lim_{x \to 0^+} f(x) = 4 - 0 = 4.$$ Since the left-hand and right-hand limits are different, $$f(x)$$ is **discontinuous at $$x = 0$$**. #### Checking $$x = 4$$: - **Left-hand limit** as $$x \to 4^-$$: $$\lim_{x \to 4^-} f(x) = 4 - 4 = 0.$$ - **Right-hand limit** as $$x \to 4^+$$: $$\lim_{x \to 4^+} f(x) = (4 - 4)^2 = 0.$$ Since the left-hand and right-hand limits are equal and equal to $$f(4) = 0$$, the function is **continuous at $$x = 4$$**. Thus, $$f(x)$$ is only discontinuous at $$x = 0$$. ### Part (b): Determine whether $$f$$ is continuous from the right, from the left, or neither at $$x = 0$$. - From part (a), we found that the left-hand limit at $$x = 0$$ is 2, and the right-hand limit is 4. Since these limits do not match, $$f(x)$$ is neither continuous from the right nor from the left. The correct answer should be "**neither**." It seems the selected option in the image ("continuous from the right") is incorrect. --- Would you like more details on the continuity concepts or any other part? Here are a few related questions you might find helpful: 1. How do you determine the left-hand and right-hand limits for piecewise functions? 2. What does it mean for a function to be continuous from the left or the right? 3. Can a function be discontinuous at one point but still continuous elsewhere? 4. What types of discontinuities can a function have (e.g., jump, infinite, removable)? 5. How can you graphically represent the continuity of a piecewise function? **Tip:** When working with piecewise functions, always check for continuity at the points where the pieces switch by comparing the left-hand and right-hand limits.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Formulas
Left-hand limit: lim_{x -> c^-} f(x)
Right-hand limit: lim_{x -> c^+} f(x)
Theorems
A function is continuous at x = c if lim_{x -> c^-} f(x) = lim_{x -> c^+} f(x) = f(c).
If the left-hand and right-hand limits at a point differ, the function is discontinuous at that point.
Suitable Grade Level
Grades 10-12