For which value of c is the function f(x) = { x^2, x < c; 2x + 1, x >= c c

For which value of c is the function f(x) = { x^2, x < c; 2x + 1, x >= c continuous at x = c?

For which value of c is the function f(x) = { x^2, x < c; 2x + 1, x >= c continuous at x = c?

Step 1: Understand that we need to find a value of c where the function f(x) is continuous at x = c.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the two parts of the function: f(x) = x^2 for x < c and f(x) = 2x + 1 for x >= c.
Step 4: Set the two parts equal to each other at x = c: x^2 = 2x + 1.
Step 5: Substitute x with c in the equation: c^2 = 2c + 1.
Step 6: Rearrange the equation to form a standard quadratic equation: c^2 - 2c - 1 = 0.
Step 7: Solve the quadratic equation using the quadratic formula or factoring. In this case, we can find that c = 2 is a solution.
Step 8: Verify that c = 2 makes both parts of the function equal at x = c.

Continuity of Piecewise Functions – The question tests the understanding of how to determine the continuity of a piecewise function at a specific point by ensuring the left-hand limit equals the right-hand limit at that point.
Solving Equations – The question requires solving an equation derived from setting the two pieces of the function equal to each other at the point of interest.