To graph the piecewise functions from the image, I'll break down each function and provide instructions on how to graph them. Let's go through them one by one.

1) $f(x) = \begin{cases} x^2 + 2 & \text{if } x < 1 \\ 3x - 5 & \text{if } 1 \leq x \leq 3 \end{cases}$

Instructions to Graph:

• For $x < 1$:

  • The function is $f(x) = x^2 + 2$, a parabolic curve shifted upwards by 2 units. Plot points for values like $x = 0, -1, -2$, and so on. Do not include $x = 1$.
  • Since this is a strict inequality, use an open circle at $x = 1$.

• For $1 \leq x \leq 3$:

  • The function is $f(x) = 3x - 5$, a linear function with a slope of 3 and y-intercept at -5.
  • Plot points for $x = 1, 2, 3$, and connect them with a straight line. Since the inequality is $\leq$, use closed circles at $x = 1$ and $x = 3$.

2) $f(x) = \begin{cases} x^2 & \text{if } x \leq 0 \\ x + 1 & \text{if } x > 0 \end{cases}$

Instructions to Graph:

• For $x \leq 0$:

  • The function is $f(x) = x^2$, a parabola opening upwards. Plot points like $x = 0, -1, -2$, and use a closed circle at $x = 0$.

• For $x > 0$:

  • The function is $f(x) = x + 1$, a linear function with slope 1 and y-intercept at 1.
  • Plot points starting from $x = 0$ but use an open circle at $x = 0$ because $x > 0$, and plot more points for $x > 0$.

3) $f(x) = \begin{cases} 2x + 3 & \text{if } x < 1 \\ x - 1 & \text{if } 1 \leq x \leq 6 \end{cases}$

Instructions to Graph:

• For $x < 1$:

  • The function is $f(x) = 2x + 3$, a linear function with slope 2 and y-intercept at 3.
  • Plot points like $x = 0, -1$, and use an open circle at $x = 1$.

• For $1 \leq x \leq 6$:

  • The function is $f(x) = x - 1$, a linear function with slope 1 and y-intercept at -1.
  • Plot points for $x = 1, 2, 3, 4, 5, 6$, and connect them with a straight line. Use closed circles at $x = 1$ and $x = 6$.

Let me know if you'd like a detailed breakdown of how to plot these graphs or any clarification!

Related Questions:

1. How do you determine where to place open and closed circles on piecewise functions?
2. Can a piecewise function have both a quadratic and a linear part?
3. What are the key differences in graphing strict inequalities ($<$ or $>$) vs non-strict inequalities ($\leq$ or $\geq$)?
4. How do you evaluate a piecewise function at a point where the definition changes?
5. How would the graph change if the second part of the function had a cubic term?

Tip:

When graphing piecewise functions, always pay attention to the type of inequality to know whether to use open or closed circles at the boundary points!