Determine the continuity of f(x) = { x^2 - 1, x < 1; 3, x = 1; 2x, x > 1 }
Practice Questions
Q1
Determine the continuity of f(x) = { x^2 - 1, x < 1; 3, x = 1; 2x, x > 1 } at x = 1.
Continuous
Discontinuous
Depends on x
Not defined
Questions & Step-by-Step Solutions
Determine the continuity of f(x) = { x^2 - 1, x < 1; 3, x = 1; 2x, x > 1 } at x = 1.
Step 1: Identify the function f(x) and its pieces: f(x) = x^2 - 1 for x < 1, f(x) = 3 for x = 1, and f(x) = 2x for x > 1.
Step 2: Calculate the left limit as x approaches 1 from the left (x < 1). This means we use the piece x^2 - 1. So, we find the limit: lim (x -> 1-) f(x) = 1^2 - 1 = 0.
Step 3: Calculate the right limit as x approaches 1 from the right (x > 1). This means we use the piece 2x. So, we find the limit: lim (x -> 1+) f(x) = 2 * 1 = 2.
Step 4: Find the value of the function at x = 1. This is given as f(1) = 3.
Step 5: Compare the left limit, right limit, and the value of the function at x = 1. We have left limit = 0, right limit = 2, and f(1) = 3.
Step 6: Since the left limit (0) does not equal the right limit (2) and f(1) (3) is also different, the function is discontinuous at x = 1.
Piecewise Functions – Understanding how to evaluate functions defined by different expressions based on the input value.
Limits – Calculating left-hand and right-hand limits to determine continuity at a specific point.
Continuity – A function is continuous at a point if the left limit, right limit, and function value at that point are all equal.
