If f(x) = { x^2, x < 3; k, x = 3; 3x - 2, x > 3 } is continuous at x = 3,
Practice Questions
Q1
If f(x) = { x^2, x < 3; k, x = 3; 3x - 2, x > 3 } is continuous at x = 3, what is k?
7
9
8
6
Questions & Step-by-Step Solutions
If f(x) = { x^2, x < 3; k, x = 3; 3x - 2, x > 3 } is continuous at x = 3, what is k?
Step 1: Understand that for a function to be continuous at a point, the value of the function at that point must equal the limit of the function as it approaches that point from both sides.
Step 2: Identify the function f(x) and the point of interest, which is x = 3.
Step 3: Calculate the limit of f(x) as x approaches 3 from the left (x < 3). This is given by the expression x^2.
Step 4: Substitute 3 into the expression x^2: 3^2 = 9. So, the limit from the left is 9.
Step 5: Calculate the limit of f(x) as x approaches 3 from the right (x > 3). This is given by the expression 3x - 2.
Step 6: Substitute 3 into the expression 3x - 2: 3(3) - 2 = 9 - 2 = 7. So, the limit from the right is 7.
Step 7: For the function to be continuous at x = 3, the value of the function at x = 3 (which is k) must equal both limits. Since the left limit is 9 and the right limit is 7, we need to set k equal to the left limit.
Step 8: Therefore, k must be 9 for the function to be continuous at x = 3.
Continuity of Functions – Understanding the conditions under which a piecewise function is continuous at a point, specifically the requirement that the left-hand limit, right-hand limit, and the function value at that point must all be equal.
Limits – Calculating the limits of a function as it approaches a specific point from both sides.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value.
