If f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2, what
Practice Questions
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If f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2, what is the value of f(2)?
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Questions & Step-by-Step Solutions
If f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2, what is the value of f(2)?
Step 1: Understand that f(x) is a piecewise function, meaning it has different rules for different values of x.
Step 2: Identify the three parts of the function: f(x) = x^2 when x < 2, f(x) = 4 when x = 2, and f(x) = 2x when x > 2.
Step 3: To check for continuity at x = 2, we need to find the limit of f(x) as x approaches 2 from both sides.
Step 4: Calculate the limit as x approaches 2 from the left (x < 2). This means using f(x) = x^2. So, limit as x approaches 2 from the left is 2^2 = 4.
Step 5: Calculate the limit as x approaches 2 from the right (x > 2). This means using f(x) = 2x. So, limit as x approaches 2 from the right is 2*2 = 4.
Step 6: Since both limits (from the left and right) equal 4, we conclude that for f(x) to be continuous at x = 2, f(2) must also equal 4.
Step 7: Therefore, the value of f(2) is 4.
Continuity of Functions – Understanding the definition of continuity at a point, which requires that the function's value at that point equals the limit of the function as it approaches that point from both sides.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly how to evaluate them at specific points.
Limits – Calculating the limit of a function as it approaches a certain value from both the left and right sides.
