Math Problem Statement
Suppose that the function f is defined, for all real numbers, as follows: f(x) = { x - 4 if x < 2, -2x + 2 if x ≥ 2. Graph the function f. Then determine whether or not the function is continuous.
Solution
The piecewise function given is:
\begin{cases} x - 4 & \text{if } x < 2, \\ -2x + 2 & \text{if } x \geq 2. \end{cases}$$ ### 1. Graph of the Function: To graph the function, let's consider each piece separately: 1. **For $$x < 2$$:** The function is $$f(x) = x - 4$$. This is a linear function with a slope of 1 and y-intercept at $$-4$$. 2. **For $$x \geq 2$$:** The function is $$f(x) = -2x + 2$$. This is also a linear function with a slope of $$-2$$ and y-intercept at 2. ### 2. Continuity Check: To determine whether the function is continuous at $$x = 2$$, we need to check: - **Limit from the left ($$x \to 2^-$$)**: $$\lim_{x \to 2^-} f(x) = 2 - 4 = -2.$$ - **Limit from the right ($$x \to 2^+$$)**: $$\lim_{x \to 2^+} f(x) = -2(2) + 2 = -2.$$ - **Value of the function at $$x = 2$$**: $$f(2) = -2(2) + 2 = -2.$$ Since the left-hand limit, right-hand limit, and the value of the function at $$x = 2$$ are all equal, $$f(x)$$ is continuous at $$x = 2$$. Thus, the function $$f(x)$$ is continuous everywhere. Would you like me to graph the function or provide more details? ### Related Questions: 1. What is the slope of each piece of the function? 2. How do you determine the continuity of a piecewise function? 3. How would the graph change if one of the inequalities were strict? 4. Can a function be discontinuous even if the left-hand and right-hand limits are equal? 5. What are the domain and range of this piecewise function? ### Tip: For piecewise functions, always check the function's value and limits at the boundary points to determine continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Graphing Functions
Limits
Formulas
f(x) = x - 4 if x < 2
f(x) = -2x + 2 if x ≥ 2
Limit from the left: lim x -> 2- (x - 4)
Limit from the right: lim x -> 2+ (-2x + 2)
Theorems
Limit Theorem
Definition of Continuity
Suitable Grade Level
Grades 10-12