Find the value of m for which the function f(x) = { 2x + m, x < 1; x^2 + 1, x
Practice Questions
Q1
Find the value of m for which the function f(x) = { 2x + m, x < 1; x^2 + 1, x >= 1 is continuous at x = 1.
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Questions & Step-by-Step Solutions
Find the value of m for which the function f(x) = { 2x + m, x < 1; x^2 + 1, x >= 1 is continuous at x = 1.
Step 1: Understand that we need to find the value of m that makes the function f(x) continuous at x = 1.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the two parts of the function: f(x) = 2x + m for x < 1 and f(x) = x^2 + 1 for x >= 1.
Step 4: Calculate the left-hand limit as x approaches 1 from the left (x < 1): f(1) = 2(1) + m = 2 + m.
Step 5: Calculate the right-hand limit as x approaches 1 from the right (x >= 1): f(1) = 1^2 + 1 = 1 + 1 = 2.
Step 6: Set the left-hand limit equal to the right-hand limit: 2 + m = 2.
Step 7: Solve for m: Subtract 2 from both sides to get m = 0.
Continuity of Piecewise Functions – The question tests the understanding of continuity at a point for piecewise functions by ensuring the left-hand limit equals the right-hand limit at that point.
