Determine whether the function f(x) is continuous at x=-3. f(x)={[18-4x^(2)",",x -3],[-9+3x",",x

Question

Determine whether the function  f(x) is continuous at  x=-3.  f(x)={[18-4x^(2)",",x > -3],[-9+3x",",x <= -3]:}  f(x) is discontinuous at  x=-3  f(x) is continuous at  x=-3

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Check Function Definition: To determine if the function f(x) is continuous at x=-3, we need to check if the following three conditions are met: 1. The function is defined at x=-3. 2. The limit of f(x) as x approaches -3 from the left is equal to the limit of f(x) as x approaches -3 from the right. 3. The limit of f(x) as x approaches -3 is equal to the function value at x=-3.Find Left Limit: First, let's check if the function is defined at x=-3. We look at the piece of the function that applies when x <= -3, which is f(x) = -9 + 3x. Plugging in x=-3, we get f(-3) = -9 + 3(-3) = -9 - 9 = -18. So, the function is defined at x=-3.Find Right Limit: Next, we need to find the limit of f(x) as x approaches -3 from the left. Since x is approaching -3 from the left (values less than -3), we use the piece of the function defined for x <= -3, which is f(x) = -9 + 3x. The limit as x approaches -3 from the left is lim(x->-3-) f(x) = -9 + 3(-3) = -9 - 9 = -18.Compare Limits: Now, we need to find the limit of f(x) as x approaches -3 from the right. Since x is approaching -3 from the right (values greater than -3), we use the piece of the function defined for x > -3, which is f(x) = 18 - 4x^2. The limit as x approaches -3 from the right is lim(x->-3+) f(x) = 18 - 4(-3)^2 = 18 - 4(9) = 18 - 36 = -18.Compare Limit and Function Value: We have found that the limit from the left is -18 and the limit from the right is also -18. Since these two limits are equal, the second condition for continuity is met.Verify Continuity: Finally, we compare the limit of f(x) as x approaches -3 to the function value at x=-3. We have already calculated that the limit from both sides is -18 and the function value at x=-3 is also -18. Since these are equal, the third condition for continuity is met.All three conditions for continuity at x=-3 are met: the function is defined at x=-3, the limits from both sides are equal, and the limit equals the function value at x=-3. Therefore, f(x) is continuous at x=-3.

Determine whether the function $f(x)$ is continuous at $x=-3$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 18-4 x^{2}, & x>-3 \\ -9+3 x, & x \leq-3 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=-3$ $\newline$ $f(x)$ is continuous at $x=-3$

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