Determine the value of c for which the function f(x) = { 3x + c, x < 1; 2x^2
Practice Questions
Q1
Determine the value of c for which the function f(x) = { 3x + c, x < 1; 2x^2 - 1, x >= 1 } is continuous at x = 1.
-1
0
1
2
Questions & Step-by-Step Solutions
Determine the value of c for which the function f(x) = { 3x + c, x < 1; 2x^2 - 1, x >= 1 } is continuous at x = 1.
Correct Answer: -2
Step 1: Identify the two pieces of the function f(x). The first piece is 3x + c for x < 1, and the second piece is 2x^2 - 1 for x >= 1.
Step 2: Find the value of the first piece when x is equal to 1. Substitute x = 1 into the first piece: 3(1) + c = 3 + c.
Step 3: Find the value of the second piece when x is equal to 1. Substitute x = 1 into the second piece: 2(1)^2 - 1 = 2 - 1 = 1.
Step 4: Set the two results from Step 2 and Step 3 equal to each other to ensure continuity at x = 1: 3 + c = 1.
Step 5: Solve for c by isolating it: c = 1 - 3.
Step 6: Calculate the value of c: c = -2.
Piecewise Functions – Understanding how to evaluate and ensure continuity at a point where a piecewise function changes its definition.
Continuity – The concept that a function is continuous at a point if the limit from the left equals the limit from the right and equals the function's value at that point.
Solving Equations – Setting the two pieces of the function equal to each other to find the necessary condition for continuity.
