Determine if the function f(x) = { x^2, x < 1; x + 1, x >= 1 } is continuo
Practice Questions
Q1
Determine if the function f(x) = { x^2, x < 1; x + 1, x >= 1 } is continuous at x = 1.
Yes
No
Depends on x
None of the above
Questions & Step-by-Step Solutions
Determine if the function f(x) = { x^2, x < 1; x + 1, x >= 1 } is continuous at x = 1.
Correct Answer: Yes, the function is continuous at x = 1.
Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 when x < 1 and f(x) = x + 1 when x >= 1.
Step 2: Find the value of f(1) using the second part of the function since 1 is included in that part. Calculate f(1) = 1 + 1 = 2.
Step 3: Calculate the limit of f(x) as x approaches 1 from the left (x < 1). Use the first part of the function: limit as x approaches 1 from the left of f(x) = x^2 = 1^2 = 1.
Step 4: Calculate the limit of f(x) as x approaches 1 from the right (x >= 1). Use the second part of the function: limit as x approaches 1 from the right of f(x) = x + 1 = 1 + 1 = 2.
Step 5: Compare the left-hand limit (1) and the right-hand limit (2). Since they are not equal, the function is not continuous at x = 1.
Continuity of Functions – Understanding the definition of continuity at a point, which requires that the limit from the left equals the limit from the right and both equal the function's value at that point.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly at the point where the definition changes.
Limits – Calculating the left-hand limit and right-hand limit at a specific point to determine continuity.
