Math Problem Statement
Suppose that the function f is defined for all real numbers as follows: f(x) = -5x + 3 if x ≤ 2 and f(x) = x - 4 if x > 2. Graph the function and determine whether or not it is continuous.
Solution
We are given a piecewise function defined as:
-5x + 3 & \text{if } x \leq 2 \\ x - 4 & \text{if } x > 2 \end{cases}$$ ### Step 1: Graph the function We need to plot the two segments of the piecewise function. - **For $$x \leq 2$$**, the equation is $$f(x) = -5x + 3$$. - The slope is -5, and the y-intercept is 3. - At $$x = 2$$, $$f(2) = -5(2) + 3 = -7$$. So, this point is $$(2, -7)$$. - **For $$x > 2$$**, the equation is $$f(x) = x - 4$$. - The slope is 1, and the y-intercept is -4. - At $$x = 2$$, this part of the function is not defined, but at $$x = 2^+$$, the limit would approach $$2 - 4 = -2$$. ### Step 2: Continuity Check For a function to be continuous at $$x = 2$$, three conditions must be satisfied: 1. $$f(2)$$ must exist. 2. The left-hand limit $$\lim_{x \to 2^-} f(x)$$ must exist. 3. The right-hand limit $$\lim_{x \to 2^+} f(x)$$ must exist, and both limits must be equal. - $$f(2) = -7$$ (from the first piece). - Left-hand limit $$\lim_{x \to 2^-} (-5x + 3) = -7$$. - Right-hand limit $$\lim_{x \to 2^+} (x - 4) = -2$$. Since the left-hand limit (-7) and the right-hand limit (-2) are not equal, the function is **not continuous at $$x = 2$$**. Would you like more details on any of the steps or graph plotting? Here are some related questions for further practice: 1. What are the conditions for a function to be continuous at a point? 2. How do you compute the limits from the left and right for a piecewise function? 3. Can you have continuity at a point where the function is not defined? 4. What is the graphical interpretation of a discontinuity in a piecewise function? 5. How can you modify the given piecewise function to make it continuous? **Tip**: Always verify both left and right limits when checking for continuity at a point!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
Left-hand limit and right-hand limit
Definition of continuity at a point
Theorems
Continuity at a Point Theorem
Limit Definition for Continuity
Suitable Grade Level
Grades 10-12