Find the value of c such that the function f(x) = { x^2 + c, x < 2; 4, x >
Practice Questions
Q1
Find the value of c such that the function f(x) = { x^2 + c, x < 2; 4, x >= 2 } is continuous at x = 2.
0
2
4
6
Questions & Step-by-Step Solutions
Find the value of c such that the function f(x) = { x^2 + c, x < 2; 4, x >= 2 } is continuous at x = 2.
Correct Answer: 0
Step 1: Understand that we want the function f(x) to be continuous at x = 2.
Step 2: Recognize that for f(x) to be continuous at x = 2, the value of f(x) when approaching from the left (x < 2) must equal the value of f(x) when approaching from the right (x >= 2).
Step 3: Write down the two pieces of the function: f(x) = x^2 + c for x < 2 and f(x) = 4 for x >= 2.
Step 4: Calculate the value of f(x) as x approaches 2 from the left: f(2) = 2^2 + c = 4 + c.
Step 5: Calculate the value of f(x) as x approaches 2 from the right: f(2) = 4.
Step 6: Set the two values equal to each other: 4 + c = 4.
Step 7: Solve for c by subtracting 4 from both sides: c = 4 - 4.
Step 8: Simplify the equation to find c: c = 0.
Continuity of Functions – Understanding how to ensure a piecewise function is continuous at a specific point by equating the limits from both sides.
Piecewise Functions – Recognizing how to handle different expressions for different intervals of the independent variable.
