Math Problem Statement
Graph the piecewise function f(x) and determine whether or not it is continuous. The function is defined as follows: f(x) = {x^2 + 1 if x < 1, 2 if x = 1, 8 - 3x if x > 1}.
Solution
The problem involves analyzing a piecewise-defined function :
\begin{cases} x^2 + 1 & \text{if } x < 1, \\ 2 & \text{if } x = 1, \\ 8 - 3x & \text{if } x > 1.$$ ### Step 1: Check Continuity To determine whether $$f(x)$$ is continuous, we examine its behavior at $$x = 1$$, the point where the definition of the function changes. #### (a) Left-hand Limit ($$\lim_{x \to 1^-} f(x)$$): For $$x < 1$$, $$f(x) = x^2 + 1$$. $$\lim_{x \to 1^-} f(x) = 1^2 + 1 = 2.$$ #### (b) Right-hand Limit ($$\lim_{x \to 1^+} f(x)$$): For $$x > 1$$, $$f(x) = 8 - 3x$$. $$\lim_{x \to 1^+} f(x) = 8 - 3(1) = 5.$$ #### (c) Value of $$f(1)$$: From the definition, $$f(1) = 2$$. #### Comparison of Limits and Function Value: - The left-hand limit ($$2$$) matches the value of $$f(1)$$. - However, the right-hand limit ($$5$$) does not match the left-hand limit. ### Conclusion: Since the left-hand limit does not equal the right-hand limit, $$f(x)$$ is **not continuous** at $$x = 1$$. ### Graph Explanation: - For $$x < 1$$, the graph is a parabola $$y = x^2 + 1$$. - At $$x = 1$$, there is a point $$(1, 2)$$. - For $$x > 1$$, the graph is a line $$y = 8 - 3x$$. Would you like me to further assist with graphing details or additional breakdowns? --- ### Related Questions: 1. What is the precise definition of continuity for a function at a point? 2. Can a function be continuous at some points but not others? 3. How do you compute limits for piecewise-defined functions? 4. How does the graph of $$y = x^2 + 1$$ differ from $$y = 8 - 3x$$? 5. What is the significance of left-hand and right-hand limits? --- ### Tip: When analyzing piecewise functions, always check for continuity by comparing the left-hand and right-hand limits with the actual value of the function at critical points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity of Functions
Limits
Formulas
Left-hand limit: lim_{x → c^-} f(x)
Right-hand limit: lim_{x → c^+} f(x)
Theorems
Continuity Definition at a Point
Suitable Grade Level
Grades 10-12
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