1. The problem asks to graph piecewise functions and determine a piecewise formula from a graph. 2. For each piecewise function, the formula is given with domain restrictions. We will restate each function clearly. 3. (a) Function: $$f(x) = \begin{cases} -\frac{3}{4}x + 2 & x \leq 0 \\ \frac{3}{2}x - 3 & x > 0 \end{cases}$$ This means for $x$ values less than or equal to 0, use the line $y = -\frac{3}{4}x + 2$, and for $x$ values greater than 0, use $y = \frac{3}{2}x - 3$. 4. (b) Function: $$f(x) = \begin{cases} -x + 6 & x < 4 \\ x - 2 & x \geq 4 \end{cases}$$ For $x$ less than 4, use $y = -x + 6$, and for $x$ greater or equal to 4, use $y = x - 2$. 5. (c) Function: $$f(x) = \begin{cases} \frac{1}{2}x + 7 & -4 \leq x \leq 2 \\ -2x + 9 & 2 < x \leq 6 \end{cases}$$ For $x$ between -4 and 2 inclusive, use $y = \frac{1}{2}x + 7$, and for $x$ between 2 and 6 (not including 2, including 6), use $y = -2x + 9$. 6. (d) Function: $$f(x) = \begin{cases} x + 3 & -3 \leq x < 3 \\ x - 3 & 3 \leq x \leq 9 \end{cases}$$ For $x$ between -3 and 3 (including -3, excluding 3), use $y = x + 3$, and for $x$ between 3 and 9 inclusive, use $y = x - 3$. 7. For question 5, we analyze the graph described: two line segments meeting at a minimum point near $(2,1)$. - The first segment decreases from approximately $( -4, 7 )$ to $( 2, 1 )$. - The second segment increases from $( 2, 1 )$ to approximately $( 8, 7 )$. 8. Find the equation of the first line segment: Slope $m_1 = \frac{1 - 7}{2 - (-4)} = \frac{-6}{6} = -1$ Using point-slope form with point $(2,1)$: $$y - 1 = -1(x - 2) \implies y = -x + 3$$ Domain: $-4 \leq x \leq 2$ 9. Find the equation of the second line segment: Slope $m_2 = \frac{7 - 1}{8 - 2} = \frac{6}{6} = 1$ Using point-slope form with point $(2,1)$: $$y - 1 = 1(x - 2) \implies y = x - 1$$ Domain: $2 < x \leq 8$ 10. Therefore, the piecewise formula for question 5 is: $$f(x) = \begin{cases} -x + 3 & -4 \leq x \leq 2 \\ x - 1 & 2 < x \leq 8 \end{cases}$$ This completes the problem.