Math Problem Statement
Gambarlah grafik fungsi berikut g(x)= -x+1 ,x < 1 g(x)= x-1, 1 < x < 2 g(x)= 5-x^2 , x >= 2 Kemudian carilah nilai (jika ada) dari (a) lim x mendekati 1 g(x) (b) g(1) (c) lim x mendekati 2- g(x) (d) lim x mendekati 2+ g(x)
Solution
The problem you provided involves a piecewise function defined as follows:
\begin{cases} -x + 1 & \text{for } x < 1 \\ x - 1 & \text{for } 1 < x < 2 \\ 5 - x^2 & \text{for } x \geq 2 \end{cases}$$ We are asked to graph the function and then solve the following questions: ### (a) $$\lim_{x \to 1} g(x)$$ This limit evaluates the behavior of the function as $$x$$ approaches 1 from both sides. We need to check: - As $$x \to 1^-$$, the function $$g(x) = -x + 1$$, so: $$\lim_{x \to 1^-} g(x) = -(1) + 1 = 0$$ - As $$x \to 1^+$$, the function $$g(x) = x - 1$$, so: $$\lim_{x \to 1^+} g(x) = (1) - 1 = 0$$ Since both one-sided limits are equal, the limit exists and is: $$\lim_{x \to 1} g(x) = 0$$ ### (b) $$g(1)$$ To find $$g(1)$$, we check the function definition. There is no explicit value for $$x = 1$$ in the piecewise function, meaning $$g(1)$$ is undefined. ### (c) $$\lim_{x \to 2^-} g(x)$$ For $$x \to 2^-$$, we use the second part of the piecewise function $$g(x) = x - 1$$, so: $$\lim_{x \to 2^-} g(x) = 2 - 1 = 1$$ ### (d) $$\lim_{x \to 2^+} g(x)$$ For $$x \to 2^+$$, we use the third part of the piecewise function $$g(x) = 5 - x^2$$, so: $$\lim_{x \to 2^+} g(x) = 5 - (2)^2 = 5 - 4 = 1$$ ### Conclusion - $$\lim_{x \to 1} g(x) = 0$$ - $$g(1)$$ is undefined. - $$\lim_{x \to 2^-} g(x) = 1$$ - $$\lim_{x \to 2^+} g(x) = 1$$ Thus, the function is continuous at $$x = 2$$, but there is a discontinuity at $$x = 1$$ because $$g(1)$$ is not defined. Would you like me to plot the graph of the function or explain any part in further detail? Here are some related questions to deepen understanding: 1. How can we determine continuity from a piecewise function? 2. What is the difference between a one-sided limit and a two-sided limit? 3. Can the function be made continuous at $$x = 1$$? 4. How would the graph behave near $$x = 1$$ and $$x = 2$$? 5. What happens if the limit from one side does not equal the other? **Tip:** When working with piecewise functions, always check both one-sided limits to determine if a function is continuous at a given point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Function
Limits
Continuity
Formulas
g(x) = -x + 1 for x < 1
g(x) = x - 1 for 1 < x < 2
g(x) = 5 - x^2 for x >= 2
Theorems
Limit Theorem
Continuity Definition
Suitable Grade Level
Grades 10-12