Math Problem Statement
Find constants a and b such that the function f(x) = {21, x ≤ -9; ax + b, -9 < x < 5; -21, x ≥ 5} is continuous on the entire real line.
Solution
To solve this problem, we need to determine the constants and such that the piecewise function is continuous over the entire real line.
The piecewise function is:
\begin{cases} 21, & x \leq -9 \\ ax + b, & -9 < x < 5 \\ -21, & x \geq 5 \end{cases}$$ ### Conditions for continuity: 1. The function must be continuous at $$x = -9$$: - The value from the first piece $$(x \leq -9)$$ must equal the value from the second piece $$(-9 < x < 5)$$ when $$x = -9$$: $$21 = a(-9) + b.$$ 2. The function must be continuous at $$x = 5$$: - The value from the second piece $$(-9 < x < 5)$$ must equal the value from the third piece $$(x \geq 5)$$ when $$x = 5$$: $$a(5) + b = -21.$$ ### Step-by-step solution: #### 1. Use the condition at $$x = -9$$: $$21 = a(-9) + b \implies -9a + b = 21.$$ This is our first equation: $$-9a + b = 21. \tag{1}$$ #### 2. Use the condition at $$x = 5$$: $$a(5) + b = -21 \implies 5a + b = -21.$$ This is our second equation: $$5a + b = -21. \tag{2}$$ #### 3. Solve the system of equations: From equation (1): $$b = 9a + 21. \tag{3}$$ Substitute $$b = 9a + 21$$ into equation (2): $$5a + (9a + 21) = -21.$$ Simplify: $$14a + 21 = -21.$$ Solve for $$a$$: $$14a = -42 \implies a = -3.$$ Substitute $$a = -3$$ into equation (3) to find $$b$$: $$b = 9(-3) + 21 = -27 + 21 = -6.$$ ### Final Answer: The constants are: $$a = -3, \, b = -6.$$ The correct choice is: $$\boxed{a = -3, \, b = -6}.$$ --- Would you like further details or have additional questions? ### Related Questions: 1. What is the definition of continuity for a piecewise function? 2. How do you solve a system of linear equations like this one? 3. Why does $$f(x)$$ need to be continuous at the boundaries? 4. How would the solution change if the piecewise conditions changed? 5. What would happen if the function was not continuous at $$x = -9$$ or $$x = 5$$? ### Tip: Always ensure the continuity of a piecewise function by matching values at the boundaries of each piece!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of Functions
Piecewise Functions
Linear Equations
Formulas
Continuity at a point: f(c-) = f(c) = f(c+)
Solving linear equations: ax + b = c
Theorems
Definition of Continuity
Suitable Grade Level
Grades 10-12