\( f(x)=\left\{\begin{array}{ll}x+2 & \text { if } x<2 \\ x+1 & \text { if } x \geq 2\end{array}\right. \)

The function \( f(x) \) is defined piecewise, meaning it has different expressions based on the value of \( x \). Let's break it down: 1. **For \( x < 2 \)**: The function is defined as \( f(x) = x + 2 \). 2. **For \( x \geq 2 \)**: The function is defined as \( f(x) = x + 1 \). ### Analyzing the Function To understand the behavior of this function, we can analyze it in two parts: #### Part 1: When \( x < 2 \) - The function is linear with a slope of 1 and a y-intercept of 2. - As \( x \) approaches 2 from the left, \( f(x) \) approaches \( 2 + 2 = 4 \). #### Part 2: When \( x \geq 2 \) - The function is also linear with a slope of 1 but a y-intercept of 1. - At \( x = 2 \), \( f(2) = 2 + 1 = 3 \). - As \( x \) increases beyond 2, \( f(x) \) continues to increase linearly. ### Continuity Check To check if the function is continuous at \( x = 2 \), we need to see if the left-hand limit equals the right-hand limit: - Left-hand limit as \( x \) approaches 2: \[ \lim_{x \to 2^-} f(x) = 4 \] - Right-hand limit as \( x \) approaches 2: \[ \lim_{x \to 2^+} f(x) = 3 \] Since \( 4 \neq 3 \), the function is not continuous at \( x = 2 \). ### Summary - The function \( f(x) \) is piecewise defined with different expressions for \( x < 2 \) and \( x \geq 2 \). - It is not continuous at \( x = 2 \) because the left-hand limit and right-hand limit do not match. If you have any specific questions or need further analysis, feel free to ask!