For what value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x &
Practice Questions
Q1
For what value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?
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Questions & Step-by-Step Solutions
For what value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Step 1: Identify the two parts of the function f(x). The first part is for x < 1: f(x) = x^3 - 3x + b. The second part is for x >= 1: f(x) = 2x + 1.
Step 2: To find the value of b that makes the function continuous at x = 1, we need to ensure that both parts of the function give the same output when x = 1.
Step 3: Calculate the output of the first part when x = 1: f(1) = 1^3 - 3(1) + b.
Step 4: Simplify the expression: f(1) = 1 - 3 + b = b - 2.
Step 5: Calculate the output of the second part when x = 1: f(1) = 2(1) + 1.
Step 7: Set the two outputs equal to each other to find b: b - 2 = 3.
Step 8: Solve for b: b = 3 + 2 = 5.
Continuity of Piecewise Functions – The question tests the understanding of how to ensure continuity at a point for piecewise functions by equating the limits from both sides.
Evaluating Limits – It requires evaluating the function from both sides of the point of interest (x = 1) to find the necessary conditions for continuity.
