Find the value of m for which the function f(x) = { 2x + m, x < 3; x^2 - 3, x >= 3 } is continuous at x = 3.
Practice Questions
1 question
Q1
Find the value of m for which the function f(x) = { 2x + m, x < 3; x^2 - 3, x >= 3 } is continuous at x = 3.
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Setting the two pieces equal at x = 3 gives us 6 + m = 6. Thus, m = 0.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the value of m for which the function f(x) = { 2x + m, x < 3; x^2 - 3, x >= 3 } is continuous at x = 3.
Solution: Setting the two pieces equal at x = 3 gives us 6 + m = 6. Thus, m = 0.
Steps: 7
Step 1: Understand that we have a piecewise function f(x) which has two parts: one for x < 3 and another for x >= 3.
Step 2: Identify the two parts of the function: f(x) = 2x + m for x < 3 and f(x) = x^2 - 3 for x >= 3.
Step 3: To find the value of m that makes the function continuous at x = 3, we need to ensure that both parts of the function give the same output when x = 3.
Step 4: Calculate the value of the second part of the function at x = 3: f(3) = 3^2 - 3 = 9 - 3 = 6.
Step 5: Now, set the first part of the function equal to this value when x approaches 3 from the left: 2(3) + m = 6.
Step 6: Simplify the equation: 6 + m = 6.
Step 7: Solve for m by subtracting 6 from both sides: m = 6 - 6 = 0.
