Questions: Suppose that the function f is defined, for all real numbers, as follows. f(x)= -x+3 if x<-1 -3x+1 if x ≥ -1 Graph the function f. Then determine whether or not the function is continuous.

Solution Steps

The function \( f(x) \) is defined as: \[ f(x)=\left\{\begin{array}{cc} -x+3 & \text { if } x<-1 \\ -3 x+1 & \text { if } x \geq-1 \end{array}\right. \]

To determine if the function is continuous at \( x = -1 \), we need to check the left-hand limit, right-hand limit, and the value of the function at \( x = -1 \).

• Left-hand limit as \( x \) approaches \(-1\): \[ \lim_{{x \to -1^-}} f(x) = -(-1) + 3 = 4 \]

• Right-hand limit as \( x \) approaches \(-1\): \[ \lim_{{x \to -1^+}} f(x) = -3(-1) + 1 = 4 \]

• Value of the function at \( x = -1 \): \[ f(-1) = -3(-1) + 1 = 4 \]

Since the left-hand limit, right-hand limit, and the value of the function at \( x = -1 \) are all equal, the function is continuous at \( x = -1 \).

Final Answer