1. **State the problems:** We need to express and graph the piecewise functions: 7. $$f(x) = \begin{cases} -3x + 1, & x \leq 1 \\ x + 1, & x > 1 \end{cases}$$ 8. $$f(x) = \begin{cases} 2x - 1, & x < 3 \\ -2x + 4, & x \geq 3 \end{cases}$$ 2. **Recall the definition of piecewise functions:** A piecewise function is defined by different expressions depending on the domain intervals. 3. **Analyze problem 7:** - For $$x \leq 1$$, the function is $$f(x) = -3x + 1$$. - For $$x > 1$$, the function is $$f(x) = x + 1$$. 4. **Graph problem 7:** - For $$x \leq 1$$, plot the line $$y = -3x + 1$$ up to and including $$x=1$$. - For $$x > 1$$, plot the line $$y = x + 1$$ starting just after $$x=1$$. - At $$x=1$$, calculate $$f(1)$$ from the first piece: $$f(1) = -3(1) + 1 = -2$$. - The point at $$x=1$$ is $$(-2)$$ on the first line, and the second line at $$x=1$$ is $$1 + 1 = 2$$, so there is a jump discontinuity. 5. **Analyze problem 8:** - For $$x < 3$$, the function is $$f(x) = 2x - 1$$. - For $$x \geq 3$$, the function is $$f(x) = -2x + 4$$. 6. **Graph problem 8:** - For $$x < 3$$, plot the line $$y = 2x - 1$$ up to but not including $$x=3$$. - For $$x \geq 3$$, plot the line $$y = -2x + 4$$ starting at $$x=3$$. - Calculate $$f(3)$$ from the second piece: $$f(3) = -2(3) + 4 = -6 + 4 = -2$$. - The first piece at $$x=3$$ is $$2(3) - 1 = 6 - 1 = 5$$, so there is a jump discontinuity at $$x=3$$. **Final piecewise functions:** 7. $$f(x) = \begin{cases} -3x + 1, & x \leq 1 \\ x + 1, & x > 1 \end{cases}$$ 8. $$f(x) = \begin{cases} 2x - 1, & x < 3 \\ -2x + 4, & x \geq 3 \end{cases}$$