Let's break down and sketch the piecewise function \( h(x) \): \[ h(x) = \begin{cases} h(-x) + 1 & \text{for } x < \\ 1 & \text{for } \leq x < 2 \\ |x-1| & \text{for } 2 \leq x < 4 \\ x^2 + 6x - 10 & \text{for } x \geq 4 \\ \frac{(x-4)^3}{3} - 2 & \text{for } x \geq 4 \end{cases} \] But notice: There are two expressions for \( x \geq 4 \), which is unusual. In such cases, let's check if perhaps one should be for \( x < 4 \). Let's proceed as written, but **normally, a function should have only one rule for each interval**. ## Step 1: Analyze Each Piece ### 1. \( h(-x) + 1 \) for \( x < \) - This means for \( x < \), the value of \( h(x) \) depends on the value of the function at the positive counterpart plus 1. ### 2. \( 1 \) for \( \leq x < 2 \) - Constant function, horizontal line at \( y = 1 \) for \( x \) in \([, 2)\). ### 3. \( |x-1| \) for \( 2 \leq x < 4 \) - Absolute value, V-shaped graph shifted right by 1, for \( x \) between 2 and 4. ### 4. \( x^2 + 6x - 10 \) for \( x \geq 4 \) - Parabola, starts at \( x = 4 \). ### 5. \( \frac{(x-4)^3}{3} - 2 \) for \( x \geq 4 \) - Cubic, starts at \( x = 4 \). ## Step 2: Find Values at Key Points ### For \( x = \): - Use \( 1 \), so \( h() = 1 \). ### For \( x = 2 \): - Use \( |x-1| \), so \( h(2) = |2-1| = 1 \). ### For \( x = 4 \): - \( x^2 + 6x - 10 = 16 + 24 - 10 = 30 \) - \( \frac{(4-4)^3}{3} - 2 = - 2 = -2 \) - **Ambiguity:** Both rules are given for \( x \geq 4 \). Normally, one would apply. ## Step 3: Sketch Each Piece ### \( \leq x < 2 \): \( h(x) = 1 \) - Draw a horizontal line at \( y = 1 \) from \( x = \) to \( x = 2 \). ### \( 2 \leq x < 4 \): \( h(x) = |x-1| \) - From \( x = 2 \) to \( x = 4 \), \( h(x) = x - 1 \), since \( x > 1 \). - At \( x = 2 \): \( h(2) = 1 \) - At \( x = 4 \): \( h(4) = 3 \) ### \( x \geq 4 \): Two rules - Parabola: At \( x = 4 \), value is 30 and grows quickly. - Cubic: At \( x = 4 \), value is -2. ### \( x < \): \( h(-x) + 1 \) - For each \( x < \), reflect about \( y \)-axis and add 1 to the value at \( -x \). #### Example: \( x = -1 \) - \( h(-(-1)) + 1 = h(1) + 1 \) - \( h(1) \) falls in \( \leq x < 2 \): \( h(1) = 1 \) - So, \( h(-1) = 1 + 1 = 2 \) #### Example: \( x = -2 \) - \( h(-2) = h(2) + 1 \) - \( h(2) = 1 \) (since \( 2 \leq x < 4 \): \( h(2) = 1 \)) - So, \( h(-2) = 2 \) #### Example: \( x = -3 \) - \( h(-3) = h(3) + 1 \) - \( h(3) = 2 \) (since \( 2 \leq x < 4 \): \( h(3) = |3-1| = 2 \)) - So, \( h(-3) = 3 \) ## Step 4: Final Sketch (Textual Description) - For \( x < \), the function mirrors the right side plus 1. - For \( \leq x < 2 \), \( h(x) = 1 \). - For \( 2 \leq x < 4 \), \( h(x) = x - 1 \). - For \( x \geq 4 \), function is ambiguous: two possible branches, one quadratic and one cubic. ### **Graph Overview** ``` x < : Reflection of right side + 1 ≤ x < 2: h(x) = 1 (horizontal line) 2 ≤ x < 4: h(x) = x - 1 (line from (2,1) to (4,3)) x ≥ 4: h(x) = x^2 + 6x - 10 (parabola, e.g. (4,30)), or h(x) = ((x-4)^3)/3 - 2 (cubic, e.g. (4,-2)) ``` ## **Final Answer (Summary Table)** | Interval | Expression | Key Values | |--------------------|--------------------------|---------------------------| | \( x < \) | \( h(-x) + 1 \) | \( h(-1) = 2, h(-2) = 2 \) | | \( \leq x < 2 \) | \( 1 \) | \( h() = 1, h(1) = 1 \) | | \( 2 \leq x < 4 \) | \( |x-1| = x - 1 \) | \( h(2) = 1, h(3) = 2 \) | | \( x \geq 4 \) | \( x^2 + 6x - 10 \) | \( h(4) = 30 \) | | \( x \geq 4 \) | \( \frac{(x-4)^3}{3} - 2 \)| \( h(4) = -2 \) | --- **Note:** Please clarify which rule to use for \( x \geq 4 \) if needed. The graph for \( x \geq 4 \) would either jump sharply up (parabola) or continue smoothly from the previous piece (cubic). --- ### If you need the sketch drawn, please let me know!

# Piecewise Function Analysis and Sketch ## Function Definition The piecewise function \( h(x) \) is defined as follows: \[ h(x) = \begin{cases} \dfrac{1}{|x-1|}, & 0 \leq x < 2, \\ -x^2 + 6x - 10, & 2 \leq x < 4, \\ (x-4)^3 - 2, & x \geq 4, \\ -h(-x), & x < 0. \end{cases} \] ## Step-by-Step Analysis ### Piece 1: \( 0 \leq x < 2 \) - **Expression:** \( h(x) = \frac{1}{|x-1|} \) - For \( 0 \leq x < 1 \): \( h(x) = \frac{1}{1-x} \) increases from \( h(0) = 1 \) to \( +\infty \) as \( x \to 1^- \). - For \( 1 < x < 2 \): \( h(x) = \frac{1}{x-1} \) decreases from \( +\infty \) to \( h(2^-) = 1 \). - **Key Points:** - **Vertical Asymptote:** At \( x = 1 \). - Value at \( x = 0 \): \( h(0) = 1 \). ### Piece 2: \( 2 \leq x < 4 \) - **Expression:** \( h(x) = -x^2 + 6x - 10 \) - This can be rewritten as \( h(x) = -(x-3)^2 - 1 \) (a downward parabola with vertex at \( (3, -1) \)). - **Key Points:** - \( h(2) = -2 \). - \( h(4^-) = -2 \). - **Jump Discontinuity:** At \( x = 2 \), left limit \( h(2^-) = 1 \) and right limit \( h(2) = -2 \). ### Piece 3: \( x \geq 4 \) - **Expression:** \( h(x) = (x-4)^3 - 2 \) - Starts at \( h(4) = -2 \) and increases without bound as \( x \) increases. - **Continuity at \( x = 4 \):** \( h(4) = -2 \) matches the limit from the left. ### Piece 4: \( x < 0 \) - **Expression:** \( h(x) = -h(-x) \) - This defines \( h(x) \) for negative values by reflecting the positive side across the origin. - **Key Points:** - \( h(0^-) = -1 \). - Asymptotes and values mirror those on the right side. ## Key Values and Limits | Points | Values | |----------------|--------------------------| | \( h(0) \) | \( 1 \) | | \( h(0^-) \) | \( -1 \) (left limit) | | \( \lim_{x \to 1^\pm} h(x) \) | \( +\infty \) (vertical asymptote) | | \( h(2) \) | \( -2 \) (jump at \( x=2 \)) | | \( h(3) \) | \( -1 \) (vertex of parabola) | | \( h(4) \) | \( -2 \) (continuous) | | \( h(-2) \) | \( 2 \) (reflection) | | \( h(-3) \) | \( 1 \) (reflection) | ## Graphing Steps 1. **Asymptotes:** Draw dashed vertical lines at \( x = 1 \) and \( x = -1 \). 2. **Plot Points:** - \( (0, 1) \) with solid dot. - \( (2, -2) \) with filled dot, and open dot at \( (2, 1) \). - \( (3, -1) \) (vertex of parabola). - \( (4, -2) \) (continuous point). 3. **Draw Functions:** - For \( 0 \leq x < 1 \): Curve upwards to infinity. - For \( 1 < x < 2 \): Curve down to \( (2, 1) \). - For \( 2 \leq x < 4 \): Downward parabola peaking at \( (3, -1) \). - For \( x \geq 4 \): Cubic function starting at \( (4, -2) \) increasing. 4. **Negatives:** Reflect the right side for \( x < 0 \), ensuring signs are flipped. ## Final Sketch Overview - **Behavior:** - Asymptotes at \( x = 1 \) and \( x = -1 \). - Jump discontinuity at \( x = 2 \). - Continuous at \( x = 4 \) but non-smooth (derivative changes). This structured breakdown allows for a clear understanding of the piecewise function and its graph.