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$Determine whether the function f(x) is continuous at x=4. f(x)={[20-x^(2)",",x <= 4],[12-2x",",x > 4]:} f(x) is continuous at x=4 f(x) is discontinuous at x=4$

Determine whether the function $f(x)$ is continuous at $x=4$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 20-x^{2}, & x \leq 4 \\ 12-2 x, & x>4 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=4$ $\newline$ $f(x)$ is discontinuous at $x=4$

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Precalculus

Identify discrete and continuous random variables

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Q. Determine whether the function

f(x)

is continuous at

x=4

\newline

f(x)=\left\{\begin{array}{ll} 20-x^{2}, & x \leq 4 \\ 12-2 x, & x>4 \end{array}\right.

\newline

f(x)

is continuous at

x=4

\newline

f(x)

is discontinuous at

x=4

Check Function Definition: To determine if the function $f(x)$ is continuous at $x=4$ , we need to check three conditions: $\newline$ $1$ . The function must be defined at $x=4$ . $\newline$ $2$ . The limit of $f(x)$ as $x$ approaches $4$ must exist. $\newline$ $3$ . The limit of $f(x)$ as $x$ approaches $4$ must be equal to the function value at $x=4$ .
Find Left Limit: First, let's check if the function is defined at $x=4$ . We have two expressions for $f(x)$ , one for $x \leq 4$ and one for $x > 4$ . We need to use the expression for $x \leq 4$ to find $f(4)$ . $\newline$ $f(4) = 20 - 4^2$ $\newline$ $f(4) = 20 - 16$ $\newline$ $f(4) = 4$ $\newline$ The function is defined at $x=4$ and $f(4) = 4$ .
Find Right Limit: Next, we need to find the limit of $f(x)$ as $x$ approaches $4$ from the left side ( $x \to 4^-$ ). We use the expression for $x \leq 4$ . $\lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} (20 - x^2)$ $\lim_{x \to 4^-} f(x) = 20 - (4)^2$ $\lim_{x \to 4^-} f(x) = 20 - 16$ $\lim_{x \to 4^-} f(x) = 4$
Verify Continuity: Now, we need to find the limit of $f(x)$ as $x$ approaches $4$ from the right side ( $x \to 4^+$ ). We use the expression for $x > 4$ . $\lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} (12 - 2x)$ $\lim_{x \to 4^+} f(x) = 12 - 2(4)$ $\lim_{x \to 4^+} f(x) = 12 - 8$ $\lim_{x \to 4^+} f(x) = 4$
Verify Continuity: Now, we need to find the limit of $f(x)$ as $x$ approaches $4$ from the right side ( $x \to 4^+$ ). We use the expression for $x > 4$ . $\lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} (12 - 2x)$ $\lim_{x \to 4^+} f(x) = 12 - 2(4)$ $\lim_{x \to 4^+} f(x) = 12 - 8$ $\lim_{x \to 4^+} f(x) = 4$ Since the limit from the left side and the limit from the right side are both equal to $4$ , and the function value at $x$ $0$ is also $4$ , all three conditions for continuity are satisfied. Therefore, $f(x)$ is continuous at $x$ $0$ .