Math Problem Statement
Graph the function f(x) = -x + 2 if x ≤ 1, and f(x) = 5x - 2 if x > 1. Then determine whether or not the function is continuous.
Solution
The function given is a piecewise function defined as:
\begin{cases} -x + 2 & \text{if } x \leq 1 \\ 5x - 2 & \text{if } x > 1 \end{cases}$$ ### Steps to Analyze the Continuity: 1. **Graphing the Function:** - For $$x \leq 1$$, the function is $$f(x) = -x + 2$$, which is a straight line with slope $$-1$$ and y-intercept at 2. - For $$x > 1$$, the function is $$f(x) = 5x - 2$$, which is a straight line with slope 5 and y-intercept $$-2$$. To graph this function: - Plot the line $$f(x) = -x + 2$$ for values of $$x \leq 1$$. This line passes through the points $$(0, 2)$$ and $$(1, 1)$$. - Plot the line $$f(x) = 5x - 2$$ for $$x > 1$$. This line passes through $$(1, 3)$$ and continues with slope 5. 2. **Determining Continuity at $$x = 1$$:** To check if the function is continuous at $$x = 1$$, we need to verify: - $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$. - **Left-hand limit** (as $$x \to 1^-$$): $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x + 2) = -(1) + 2 = 1.$$ - **Right-hand limit** (as $$x \to 1^+$$): $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (5x - 2) = 5(1) - 2 = 3.$$ - **Value of the function at $$x = 1$$**: $$f(1) = -1 + 2 = 1 \quad (\text{since we use the first piece for } x \leq 1).$$ ### Conclusion: The function is **not continuous** at $$x = 1$$, because the left-hand limit $$1$$, the right-hand limit $$3$$, and the value of the function at $$x = 1$$ are not equal. There is a jump discontinuity at $$x = 1$$. Would you like me to graph the function for you? Let me know if you need more details or clarifications! ### Questions: 1. What is the formal definition of continuity at a point? 2. Can you think of other types of discontinuities besides a jump discontinuity? 3. How do we determine if a piecewise function is differentiable at the joining point? 4. How does the slope affect the behavior of each segment of the function? 5. Can you find other points where the function might be discontinuous based on its definition? ### Tip: For piecewise functions, always check the limits from both sides at the points where the function pieces meet to determine continuity or differentiability.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Graphing Functions
Formulas
Left-hand limit: lim(x→1⁻) f(x) = lim(x→1⁻) (-x + 2)
Right-hand limit: lim(x→1⁺) f(x) = lim(x→1⁺) (5x - 2)
Theorems
Definition of Continuity
Suitable Grade Level
Grades 11-12
Related Recommendation
Graphing a Piecewise Function: f(x) with Different Linear and Constant Sections
Piecewise Function f(x) with Linear Segments
Graphing a Piecewise Function with Linear and Constant Segments
Graphing Piecewise Function and Determining Continuity
Understanding Piecewise Functions: Solve and Graph f(x) with Linear Segments