Questions: Suppose that the function f is defined, for all real numbers, as follows. f(x) = 2x-1 if x<1 -x+2 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: \[ f(x) = \begin{cases} 2x - 1 & \text{if } x < 1 \\ -x + 2 & \text{if } x \geq 1 \end{cases} \]

For \( x < 1 \), the function is \( f(x) = 2x - 1 \). This is a linear function with a slope of 2 and a y-intercept of -1. Plot this line for values of \( x \) less than 1.

For \( x \geq 1 \), the function is \( f(x) = -x + 2 \). This is a linear function with a slope of -1 and a y-intercept of 2. Plot this line for values of \( x \) greater than or equal to 1.

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

• Left-hand limit as \( x \) approaches 1: \( \lim_{{x \to 1^-}} (2x - 1) = 2(1) - 1 = 1 \)
• Right-hand limit as \( x \) approaches 1: \( \lim_{{x \to 1^+}} (-x + 2) = -1 + 2 = 1 \)
• Value of the function at \( x = 1 \): \( f(1) = -1 + 2 = 1 \)

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