What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1,

What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1, x > 1 continuous at x = 1?

What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1, x > 1 continuous at x = 1?

Step 1: Identify the function f(x) which has different expressions based on the value of x.
Step 2: Recognize that for the function to be continuous at x = 1, the value of f(1) must equal the limit of f(x) as x approaches 1.
Step 3: Calculate the limit of f(x) as x approaches 1 from the left (x < 1). This is given by the expression 2x + 1.
Step 4: Substitute x = 1 into the expression 2x + 1: 2(1) + 1 = 2 + 1 = 3.
Step 5: For continuity at x = 1, set the value of a (which is f(1)) equal to the limit found in Step 4: a = 3.
Step 6: Therefore, the value of a that makes the function continuous at x = 1 is a = 3.

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