Final answer:
To determine the values of constant k for which f(x) is continuous at x = -2, we need to use the definition of continuity and evaluate the limit of f(x) as x approaches -2 for each value of k given. The only value of k for which the function f(x) is continuous at x = -2 is k = 0.
Step-by-step explanation:
To determine the values of constant k for which f(x) is continuous at x = -2, we need to use the definition of continuity. A function f(x) is continuous at a point x = c if three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit and the value of f(x) at x = c are equal.
In this case, the function is continuous at x = -2 if and only if the limit of f(x) as x approaches -2 exists and is equal to f(-2). So, we need to evaluate the limit of f(x) as x approaches -2 for each value of k given.
a) For k = 2:
lim(x -> -2) f(x) = lim(x -> -2) (2x + 3) = 2(-2) + 3 = -1
Since the limit does not equal f(-2), the function is not continuous at x = -2.
b) For k = -2:
lim(x -> -2) f(x) = lim(x -> -2) (-2x + 3) = -2(-2) + 3 = 7
Since the limit does not equal f(-2), the function is not continuous at x = -2.
c) For k = 0:
lim(x -> -2) f(x) = lim(x -> -2) (0x + 3) = 3
Since the limit equals f(-2), the function is continuous at x = -2 for k = 0.
d) For k = 1:
lim(x -> -2) f(x) = lim(x -> -2) (x + 3) = -2 + 3 = 1
Since the limit does not equal f(-2), the function is not continuous at x = -2.
Therefore, the only value of k for which the function f(x) is continuous at x = -2 is k = 0.