Math Problem Statement
Diketahui sebuah fungsi berbentuk f(x) dengan nilai yang berbeda berdasarkan interval x, dan diberikan fungsi piecewise:
\[
f(x) =
\begin{cases}
x^2, & x < 1 \\
a, & x = 1 \\
bx + 2, & x > 1
\end{cases}
\]
Pertanyaan:
1. Tentukan nilai a dan b agar fungsi f(x) kontinu di x = 1.
2. Setelah menemukan nilai a dan b, gambar grafik kurva f(x).
Solution
Let's go through the problem step-by-step.
Given
The function is defined piecewise as:
\begin{cases} x^2, & x < 1 \\ a, & x = 1 \\ bx + 2, & x > 1 \end{cases}$$ ### Problem 1. **Find values of $$a$$ and $$b$$ that make $$f(x)$$ continuous at $$x = 1$$.** 2. **Once $$a$$ and $$b$$ are determined, plot the graph of $$f(x)$$.** ### Solution #### Part (a): Continuity at $$x = 1$$ For $$f(x)$$ to be continuous at $$x = 1$$, the left-hand limit, right-hand limit, and the function's value at $$x = 1$$ must all be equal. 1. **Left-hand limit as $$x \to 1^-$$:** $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1^2 = 1$$ 2. **Right-hand limit as $$x \to 1^+$$:** $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (bx + 2) = b(1) + 2 = b + 2$$ 3. **Value at $$x = 1$$:** $$f(1) = a$$ For continuity at $$x = 1$$, these three expressions must be equal: $$1 = a = b + 2$$ So we have two equations: 1. $$a = 1$$ 2. $$b + 2 = 1$$ Solving these: - From the first equation, $$a = 1$$. - Substitute $$a = 1$$ into the second equation: $$b + 2 = 1 \Rightarrow b = -1$$ Therefore, the values that make $$f(x)$$ continuous at $$x = 1$$ are: $$a = 1 \quad \text{and} \quad b = -1$$ #### Part (b): Graph of $$f(x)$$ With $$a = 1$$ and $$b = -1$$, we can rewrite the function $$f(x)$$ as: $$f(x) = \begin{cases} x^2, & x < 1 \\ 1, & x = 1 \\ - x + 2, & x > 1 \end{cases}$$ 1. For $$x < 1$$, the function is $$f(x) = x^2$$, which is a parabolic curve. 2. At $$x = 1$$, the function has the value $$f(1) = 1$$. 3. For $$x > 1$$, the function is $$f(x) = -x + 2$$, which is a linear function with a negative slope. Would you like me to plot this graph for you, or do you have any questions on the solution so far? --- ### Follow-up Questions 1. What does it mean for a function to be continuous at a certain point? 2. How do we determine the left-hand and right-hand limits of a piecewise function? 3. Can there be a discontinuity at $$x = 1$$ if we change the values of $$a$$ and $$b$$? 4. What happens to the graph if we change the values of $$a$$ and $$b$$ again? 5. How would you check continuity for other types of functions? #### Tip For a piecewise function, always examine each interval and boundary separately to check for continuity and differentiability.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Limit definition for continuity: \( \lim_{x \to c} f(x) = f(c) \)
Theorems
Definition of Continuity at a Point
Suitable Grade Level
Grade 11-12
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