Math Problem Statement
Suppose that the function f is defined, for all real numbers, as follows: f(x) = { 5x - 4 if x ≤ 1 ; -x + 4 if x > 1 }. Graph the function f. Then determine whether or not the function is continuous.
Solution
Let's analyze the problem presented.
The function is defined as a piecewise function:
5x - 4 & \text{if } x \leq 1 \\ -x + 4 & \text{if } x > 1 \end{cases}$$ The task involves graphing this function and determining if it is continuous. ### Steps to Determine Continuity at $$x = 1$$: 1. **Evaluate $$f(1)$$ Using the First Case:** Since $$x = 1$$ is within the first piece, $$f(1) = 5(1) - 4 = 5 - 4 = 1$$. 2. **Calculate the Left-Hand Limit as $$x \to 1^-$$:** Using $$f(x) = 5x - 4$$ as $$x \to 1^-$$, we get: $$\lim_{{x \to 1^-}} f(x) = 5(1) - 4 = 1$$ 3. **Calculate the Right-Hand Limit as $$x \to 1^+$$:** Using $$f(x) = -x + 4$$ as $$x \to 1^+$$, we get: $$\lim_{{x \to 1^+}} f(x) = -1 + 4 = 3$$ Since the left-hand limit ($$1$$) and the right-hand limit ($$3$$) are not equal, the function is **discontinuous at $$x = 1$$**. This is because, for continuity at $$x = 1$$, we would need: $$\lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1)$$ Thus, the answer to whether the function is continuous is **No**. Would you like more details or have any questions? Here are some related questions to consider: 1. How do we determine continuity for piecewise functions in general? 2. What is the importance of left-hand and right-hand limits in continuity? 3. How would the graph of $$f(x)$$ look with the discontinuity at $$x = 1$$? 4. Can we determine the type of discontinuity present here? 5. How would the continuity change if the pieces of $$f(x)$$ were adjusted to make the limits match? **Tip:** Always check both left-hand and right-hand limits when testing continuity at a point for piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity of Functions
Formulas
Left-hand limit: lim(x -> c-) f(x)
Right-hand limit: lim(x -> c+) f(x)
Theorems
Continuity at a Point
Suitable Grade Level
Grades 10-12