DETAILS MY NOTES Sketch the graph of a function \( f \) that is continuous except for the stated discontinuity. Removable discontinuity at 4, jump discontinuity at 6

Certainly! Let's delve into the details of sketching a function \( f \) that is continuous everywhere except for two specific discontinuities: 1. **Removable Discontinuity at \( x = 4 \)** 2. **Jump Discontinuity at \( x = 6 \)** ### Understanding the Discontinuities 1. **Removable Discontinuity:** - **Definition:** A point \( x = a \) is a removable discontinuity of \( f \) if \( \lim_{x \to a} f(x) \) exists, but either \( f(a) \) is not defined or \( f(a) \neq \lim_{x \to a} f(x) \). - **Graphically:** The graph has a "hole" at \( x = a \). You can "fill" this hole by redefining \( f(a) \) appropriately. 2. **Jump Discontinuity:** - **Definition:** A point \( x = b \) is a jump discontinuity of \( f \) if the left-hand limit \( \lim_{x \to b^-} f(x) \) and the right-hand limit \( \lim_{x \to b^+} f(x) \) both exist but are not equal. - **Graphically:** The graph has a "jump" at \( x = b \), showing two distinct values approaching \( b \) from the left and right. ### Constructing the Function \( f \) To create a function with the desired properties, we'll define it piecewise. Here's a step-by-step approach: 1. **Choose Intervals:** - **Interval 1:** \( x < 4 \) - **Interval 2:** \( 4 < x < 6 \) - **Interval 3:** \( x > 6 \) 2. **Define \( f \) on Each Interval:** - **For \( x < 4 \):** Let’s choose a simple linear function, e.g., \( f(x) = 2x + 1 \). - **For \( 4 < x < 6 \):** Choose another linear function, e.g., \( f(x) = -x + 10 \). - **For \( x > 6 \):** Choose yet another linear function, e.g., \( f(x) = x - 2 \). 3. **Handle the Discontinuities:** - **At \( x = 4 \) (Removable):** - **Limit from the Left:** \( \lim_{x \to 4^-} f(x) = 2(4) + 1 = 9 \). - **Limit from the Right:** \( \lim_{x \to 4^+} f(x) = -4 + 10 = 6 \). - **Correction:** To make it a removable discontinuity, both one-sided limits should be equal. Since currently they are not, let's adjust the functions to make \( \lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) \). **Revising Functions:** - **For \( x < 4 \):** Let \( f(x) = x + 5 \) (so \( f(4^-) = 9 \)). - **For \( 4 < x < 6 \):** Let \( f(x) = x + 5 \) (so \( f(4^+) = 9 \)). - **Define \( f(4) \) differently:** To create a removable discontinuity, set \( f(4) \neq 9 \). For example, \( f(4) = 7 \). - **At \( x = 6 \) (Jump):** - **Limit from the Left:** \( \lim_{x \to 6^-} f(x) = 6 + 5 = 11 \). - **Limit from the Right:** \( \lim_{x \to 6^+} f(x) = 6 - 2 = 4 \). - **Define \( f(6) \) as either 11 or 4 or any other value (doesn't affect the jump). For simplicity, let \( f(6) = 4 \). 4. **Final Piecewise Function:** \[ f(x) = \begin{cases} x + 5 & \text{if } x < 4, \\ x + 5 & \text{if } 4 < x < 6, \\ x - 2 & \text{if } x > 6, \\ 7 & \text{if } x = 4, \\ 4 & \text{if } x = 6. \end{cases} \] ### Visualizing the Graph 1. **For \( x < 4 \) and \( 4 < x < 6 \):** - The function \( f(x) = x + 5 \) is a straight line with a slope of 1 and a y-intercept at 5. - **At \( x = 4 \):** - There is a hole in the graph at \( (4, 9) \) since \( f(4) = 7 \neq 9 \). This is the **removable discontinuity**. 2. **For \( x > 6 \):** - The function \( f(x) = x - 2 \) is another straight line with a slope of 1 but a y-intercept at -2. - **At \( x = 6 \):** - The left limit is 11 (from \( f(x) = x + 5 \)) but the function jumps to 4 at \( x = 6 \). This creates the **jump discontinuity**. 3. **Connecting the Pieces:** - **Continuity except at 4 and 6:** - Between \( x < 4 \) and \( 4 < x < 6 \), the function follows \( f(x) = x + 5 \), maintaining continuity except at \( x = 4 \). - After \( x > 6 \), the function switches to \( f(x) = x - 2 \), creating a jump from the previous value of 11 to 4 at \( x = 6 \). ### Sketching the Graph While I can't provide an actual sketch here, here's how you can draw it: 1. **Plot \( f(x) = x + 5 \) for \( x < 4 \) and \( 4 < x < 6 \):** - Draw a straight line with points like \( (0, 5) \), \( (2, 7) \), \( (4, 9) \). - At \( x = 4 \), place an open circle at \( (4, 9) \) and a filled circle at \( (4, 7) \). 2. **Plot \( f(x) = x - 2 \) for \( x > 6 \):** - Draw a straight line starting from \( x > 6 \), passing through points like \( (6, 4) \), \( (8, 6) \). - At \( x = 6 \), place a filled circle at \( (6, 4) \) and an open circle at \( (6, 11) \) to indicate the jump. 3. **Discontinuities:** - **At \( x = 4 \):** A hole at \( (4, 9) \) and a point at \( (4, 7) \). - **At \( x = 6 \):** An open circle at \( (6, 11) \) and a filled circle at \( (6, 4) \). ### Summary - **Removable Discontinuity at \( x = 4 \):** The function approaches 9 from both sides but is explicitly defined as 7 at \( x = 4 \), creating a hole. - **Jump Discontinuity at \( x = 6 \):** The function jumps from approaching 11 to being 4 at \( x = 6 \). By carefully defining the function piecewise and handling the discontinuities as shown, you can effectively sketch \( f(x) \) with the desired properties.