Find each \( x \)-value at which \( f \) is discontinuous and for each \( x \)-value, determine whether \( f \) is continuous from the right, or from the left, or neither. \[ f(x)=\left\{\begin{array}{ll} x+2 & \text { if } x<1 \\ 1 / x & \text { if } 1

Ask by Ruiz May. in the United States

Feb 03,2025

To analyze the continuity of the function \( f(x) \) and sketch its graph, let's examine each piece and identify potential points of discontinuity. ### Given Function: \[ f(x)=\left\{ \begin{array}{ll} x+2 & \text{if } x < 1 \\ \frac{1}{x} & \text{if } 1 < x < 2 \\ \sqrt{x-2} & \text{if } x \geq 2 \end{array} \right. \] ### Points of Potential Discontinuity: The potential points of discontinuity are at the boundaries where the pieces of the function meet, namely at \( x = 1 \) and \( x = 2 \). --- ### 1. **At \( x = 1 \) (Smaller Value):** - **Left-Hand Limit (as \( x \to 1^- \)):** \[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x + 2) = 1 + 2 = 3 \] - **Right-Hand Limit (as \( x \to 1^+ \)):** \[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} \left( \frac{1}{x} \right) = \frac{1}{1} = 1 \] - **Function Value at \( x = 1 \):** The function \( f(x) \) is **not defined** at \( x = 1 \) based on the given piecewise definition. - **Conclusion for \( x = 1 \):** - **Continuous from the Right:** **No** - **Continuous from the Left:** **No** - **Result:** **Neither** --- ### 2. **At \( x = 2 \) (Larger Value):** - **Left-Hand Limit (as \( x \to 2^- \)):** \[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} \left( \frac{1}{x} \right) = \frac{1}{2} = 0.5 \] - **Right-Hand Limit (as \( x \to 2^+ \)):** \[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} \sqrt{x - 2} = \sqrt{2 - 2} = 0 \] - **Function Value at \( x = 2 \):** \[ f(2) = \sqrt{2 - 2} = 0 \] - **Conclusion for \( x = 2 \):** - **Continuous from the Right:** **Yes** (since \( \lim_{x \to 2^+} f(x) = f(2) \)) - **Continuous from the Left:** **No** (since \( \lim_{x \to 2^-} f(x) \neq f(2) \)) - **Result:** **Continuous from the Right** --- ### Summary of Discontinuities: | \( x \)-value | Continuous from the Right | Continuous from the Left | Neither | |---------------|---------------------------|--------------------------|---------| | \( x = 1 \) | No | No | **Yes** | | \( x = 2 \) | **Yes** | No | No | --- ### Graph of \( f(x) \): 1. **For \( x < 1 \): \( f(x) = x + 2 \)** - This is a straight line with a slope of 1 and a y-intercept at \( (0, 2) \). - As \( x \to 1^- \), \( f(x) \to 3 \). - **Open Circle at \( x = 1 \):** Since \( f(1) \) is not defined, represent with an open circle at \( (1, 3) \). 2. **For \( 1 < x < 2 \): \( f(x) = \frac{1}{x} \)** - This is a hyperbola segment decreasing from \( x = 1 \) to \( x = 2 \). - At \( x = 1 \), \( f(x) = 1 \) (but not included, so use an open circle at \( (1, 1) \)). - At \( x = 2 \), \( f(x) = 0.5 \). 3. **For \( x \geq 2 \): \( f(x) = \sqrt{x - 2} \)** - This is the right half of a parabola starting at \( (2, 0) \). - At \( x = 2 \), \( f(x) = 0 \) (use a closed circle at \( (2, 0) \)). **Visual Representation:** ``` y | 4 + | 3 + ● ← Open circle at (1,3) | / 2 +------+---------+---------+ | / \ 1 + ● √(x-2) | 0 +---------------------------- x 1 2 ``` - **Open circles** indicate points that are not included in the function. - **Closed circles** indicate points that are included in the function. - The dashed lines represent the behavior of the function approaching the discontinuities. --- ### Final Answer: - **Discontinuities at:** - \( \boxed{x=1} \) - **Neither continuous from the right nor from the left.** - \( \boxed{x=2} \) - **Continuous from the right only.** - **Graph Sketch:** - **For \( x < 1 \):** Line \( f(x) = x + 2 \) with an open circle at \( (1, 3) \). - **For \( 1 < x < 2 \):** Hyperbola \( f(x) = \frac{1}{x} \) with an open circle at \( (1, 1) \) and approaching \( (2, 0.5) \). - **For \( x \geq 2 \):** Square root function \( f(x) = \sqrt{x - 2} \) starting at \( (2, 0) \) with a closed circle.