Math Problem Statement
Solution
The problem asks to find the values of and so that the function is continuous at .
The given function is:
\begin{cases} a - x, & \text{if } x \leq 1, \\ 2x - b, & \text{if } x > 1. \end{cases}$$ For the function to be continuous at $$x = 1$$, the left-hand limit (as $$x$$ approaches 1 from the left) must equal the right-hand limit (as $$x$$ approaches 1 from the right) and must also equal the function value at $$x = 1$$. ### Steps to find $$a$$ and $$b$$: 1. **Left-hand limit** ($$x \to 1^-$$): $$\lim_{x \to 1^-} f(x) = a - 1.$$ 2. **Right-hand limit** ($$x \to 1^+$$): $$\lim_{x \to 1^+} f(x) = 2(1) - b = 2 - b.$$ 3. **Continuity condition at $$x = 1$$**: The function is continuous at $$x = 1$$ if the limits from both sides are equal: $$a - 1 = 2 - b.$$ ### Solving the equation: $$a - 1 = 2 - b.$$ Rearrange the equation: $$a + b = 3.$$ Thus, $$a$$ and $$b$$ must satisfy the equation $$a + b = 3$$. Let me know if you'd like more details on solving or verifying this, or any other questions! ### Related questions: 1. What is the definition of continuity at a point in terms of limits? 2. How can we check if a piecewise function is continuous at a given point? 3. How does the behavior of a function differ at points where it's not continuous? 4. Can the function in this problem be made differentiable at $$x = 1$$ by choosing specific values of $$a$$ and $$b$$? 5. How would the solution change if the function were defined on different intervals? ### Tip: For piecewise functions, always check continuity by ensuring the function values from both sides of the critical point are equal!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit as x approaches 1: lim_{x → 1^-} f(x) = a - 1
Right-hand limit as x approaches 1: lim_{x → 1^+} f(x) = 2(1) - b = 2 - b
Continuity condition: a - 1 = 2 - b
Theorems
Continuity at a point requires left-hand limit = right-hand limit = function value
Suitable Grade Level
High school or early college (Grades 10-12 or 1st year college)
Related Recommendation
Find values of a and b for continuity of piecewise function at x = 1
Determine Continuity of Piecewise Function at x = 1
Ensuring Continuity for a Piecewise Function at x = -2 and x = 1
Find Values of a and b for Continuity in Piecewise Function
Finding a and b for Continuity of Piecewise Function with Limits