Math Problem Statement
Determine the values of a and b so that the following function is continuous for all values of x: f(x) = { 2, x ≤ -1; ax + b, -1 < x < 3; -2, x ≥ 3 }
Solution
The problem asks to determine the values of and so that the piecewise function is continuous for all values of .
The function is given as:
\begin{cases} 2 & \text{if } x \leq -1 \\ ax + b & \text{if } -1 < x < 3 \\ -2 & \text{if } x \geq 3 \end{cases}$$ ### Continuity Condition For the function to be continuous at the boundaries $$x = -1$$ and $$x = 3$$, the function values from both sides at these points must match. 1. **At $$x = -1$$:** - The value of the function when $$x \leq -1$$ is $$f(x) = 2$$. - The value of the function when $$-1 < x < 3$$ is $$f(x) = ax + b$$. - To ensure continuity at $$x = -1$$, the two pieces must match at this point: $$a(-1) + b = 2 \quad \Rightarrow \quad -a + b = 2$$ 2. **At $$x = 3$$:** - The value of the function when $$x \geq 3$$ is $$f(x) = -2$$. - The value of the function when $$-1 < x < 3$$ is $$f(x) = ax + b$$. - To ensure continuity at $$x = 3$$, the two pieces must match at this point: $$a(3) + b = -2 \quad \Rightarrow \quad 3a + b = -2$$ ### Solving the System of Equations We now solve the system of equations: 1. $$-a + b = 2$$ 2. $$3a + b = -2$$ We can subtract equation (1) from equation (2) to eliminate $$b$$: $$(3a + b) - (-a + b) = -2 - 2$$ $$4a = -4$$ $$a = -1$$ Substitute $$a = -1$$ into equation (1): $$-(-1) + b = 2$$ $$1 + b = 2$$ $$b = 1$$ Thus, the values of $$a$$ and $$b$$ are: - $$a = -1$$ - $$b = 1$$ Would you like further explanation on any of the steps or additional assistance with related problems? Here are some related questions to explore: 1. How would you determine the continuity of a piecewise function at a different boundary? 2. What does the concept of continuity mean in the context of calculus? 3. How do you find the derivative of a piecewise function? 4. What is the difference between continuity and differentiability? 5. How can you apply this method to solve other piecewise function problems? **Tip:** Always check the continuity of piecewise functions at the transition points where the function changes its form.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Algebra
Formulas
f(x) = ax + b
Theorems
Continuity condition at transition points
Suitable Grade Level
Grades 9-12