10. [-/0.83 Points] DETAILS MY NOTES SCALCET9 2.5.045. ASK YOUR TEACHER 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 netthen. \[ r(x)=\left\{\begin{array}{ll} x+6 & \text { if } x<0 \\ e^{x} & \text { if } 0 \leq x \leq 1 \\ 7-x & \text { if } x>1 \end{array}\right. \] \( x= \) \( \square \) (smaller value) Is \( f \) continuous from the right, left, or neither at this value? continuous from the right continuous from the left neither \( x= \) \( \square \) (larger value) Is \( f \) continuous from the right, left, or neither at this value? continuous from the right continuous from the left neither Sketch the graph of \( f \).

To determine the points of discontinuity for the piecewise function \[ r(x)=\left\{\begin{array}{ll} x+6 & \text { if } x<0 \\ e^{x} & \text { if } 0 \leq x \leq 1 \\ 7-x & \text { if } x>1 \end{array}\right. \] we need to analyze the function at the boundaries of the intervals defined in the piecewise function, which are \( x = 0 \) and \( x = 1 \). ### Step 1: Check continuity at \( x = 0 \) 1. **Left-hand limit as \( x \) approaches 0**: \[ \lim_{x \to 0^-} r(x) = 0 + 6 = 6 \] 2. **Right-hand limit as \( x \) approaches 0**: \[ \lim_{x \to 0^+} r(x) = e^0 = 1 \] 3. **Value of the function at \( x = 0 \)**: \[ r(0) = e^0 = 1 \] Since the left-hand limit (6) does not equal the right-hand limit (1), \( r(x) \) is discontinuous at \( x = 0 \). ### Step 2: Check continuity at \( x = 1 \) 1. **Left-hand limit as \( x \) approaches 1**: \[ \lim_{x \to 1^-} r(x) = e^1 = e \] 2. **Right-hand limit as \( x \) approaches 1**: \[ \lim_{x \to 1^+} r(x) = 7 - 1 = 6 \] 3. **Value of the function at \( x = 1 \)**: \[ r(1) = e^1 = e \] Since the left-hand limit (e) does not equal the right-hand limit (6), \( r(x) \) is also discontinuous at \( x = 1 \). ### Summary of Discontinuities - **At \( x = 0 \)**: - Discontinuous - Continuous from the left (limit = 6) - Continuous from the right (limit = 1) - Therefore, it is neither continuous from the left nor from the right. - **At \( x = 1 \)**: - Discontinuous - Continuous from the left (limit = e) - Continuous from the right (limit = 6) - Therefore, it is neither continuous from the left nor from the right. ### Final Answers - \( x = 0 \) (smaller value) - Is \( f \) continuous from the right, left, or neither? **Neither** - \( x = 1 \) (larger value) - Is \( f \) continuous from the right, left, or neither? **Neither** ### Sketch of the Graph To sketch the graph of \( r(x) \): 1. For \( x < 0 \), the line \( y = x + 6 \) will be drawn. 2. For \( 0 \leq x \leq 1 \), the curve \( y = e^x \) will be drawn. 3. For \( x > 1 \), the line \( y = 7 - x \) will be drawn. The graph will show jumps at \( x = 0 \) and \( x = 1 \), indicating the points of discontinuity.