Math Problem Statement

Find the constants a and b such that the function

f(x)= 4, x<=negative 1; f(x)= ax+b, negative 1<x<3; f(x)=negative 4, x>=3

is continuous on the entire real line.

Solution

To ensure the function f(x)f(x) is continuous on the entire real line, the function must not have any jumps or discontinuities at the points where the piecewise function changes its definition, specifically at x=1x = -1 and x=3x = 3.

Given:

4, & x \leq -1 \\ ax + b, & -1 < x < 3 \\ -4, & x \geq 3 \end{cases}$$ ### Step 1: Continuity at $$x = -1$$ For the function to be continuous at $$x = -1$$, the left-hand limit (as $$x$$ approaches $$-1$$ from the left) must equal the right-hand limit (as $$x$$ approaches $$-1$$ from the right), and both must equal $$f(-1)$$. - Left-hand limit as $$x$$ approaches $$-1$$ from the left: $$\lim_{x \to -1^-} f(x) = 4$$ - Right-hand limit as $$x$$ approaches $$-1$$ from the right: $$\lim_{x \to -1^+} f(x) = a(-1) + b = -a + b$$ - For continuity at $$x = -1$$: $$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) = f(-1)$$ $$4 = -a + b$$ ### Step 2: Continuity at $$x = 3$$ For the function to be continuous at $$x = 3$$, the left-hand limit (as $$x$$ approaches $$3$$ from the left) must equal the right-hand limit (as $$x$$ approaches $$3$$ from the right), and both must equal $$f(3)$$. - Left-hand limit as $$x$$ approaches $$3$$ from the left: $$\lim_{x \to 3^-} f(x) = a(3) + b = 3a + b$$ - Right-hand limit as $$x$$ approaches $$3$$ from the right: $$\lim_{x \to 3^+} f(x) = -4$$ - For continuity at $$x = 3$$: $$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)$$ $$3a + b = -4$$ ### Step 3: Solve the system of equations We now have the following system of linear equations: 1. $$-a + b = 4$$ 2. $$3a + b = -4$$ Subtract equation 1 from equation 2 to eliminate $$b$$: $$(3a + b) - (-a + b) = -4 - 4$$ $$4a = -8$$ $$a = -2$$ Now, substitute $$a = -2$$ into equation 1 to solve for $$b$$: $$-a + b = 4$$ $$-(-2) + b = 4$$ $$2 + b = 4$$ $$b = 2$$ ### Final Answer The constants are $$a = -2$$ and $$b = 2$$. Would you like further details or have any questions? Here are 5 related questions you might find interesting: 1. How can you determine if a function is continuous at a specific point? 2. What is the difference between continuity and differentiability? 3. How would the answer change if the function $$f(x)$$ was defined differently for $$x \geq 3$$? 4. What are the conditions for a piecewise function to be continuous everywhere? 5. How would you approach finding the limits of more complex piecewise functions? **Tip:** When solving for continuity in piecewise functions, always equate the limits at the points where the function definition changes to ensure smooth transitions.

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Math Problem Analysis

Mathematical Concepts

Continuity
Piecewise Functions
Algebra

Formulas

Continuity condition: lim x->c- f(x) = lim x->c+ f(x) = f(c)

Theorems

Continuity at a Point

Suitable Grade Level

Grades 11-12