  ## Exercises - Restricting Relations & Piecwise-Defined Functions

1. Graph the following piecewise-defined functions. In doing so, remember to interpret more complicated "pieces" as compositions of simpler functions as helpful.

1. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} 2 &\textrm{for}& x \lt 1\\ 1 + \sqrt{x} &\textrm{for}& x \ge 1 \end{array}\right. }$

2. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} x^2 + 1 &\textrm{for}& x \lt 2\\ 5 &\textrm{for}& 2 \le x \le 4\\ 2x-5 &\textrm{for}& x \gt 4 \end{array}\right.}$

3. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} -1 &\textrm{if}& x \lt 0\\ 0 &\textrm{if}& x = 0\\ 1 &\textrm{if}& x > 0 \end{array}\right.}$

4. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} 3 &\textrm{if}& x \le -2\\ x^2 - 1 &\textrm{if}& -2 \lt x \le 3\\ -\sqrt{25-x^2} &\textrm{if}& x \gt 3 \end{array}\right.}$

5. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} -4x &,& x \le -1\\ \sqrt{1-x^2} &,& -1 \lt x \lt 1\\ -2 &,& x \ge 1 \end{array}\right.}$

6. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} 4-x^2 &,& x \lt -1\\ -2 &,& x = -1\\ \sqrt{4-x^2} &,& x \gt -1 \end{array}\right.}$

(CLICK TO SEE SOLUTION TO #1)

2. For each relation resulting from the given equation and any additional restrictions provided, solve for the indicated variable. Then graph the relation:

1. $\displaystyle{x^2 + y^2 = 16, \ \ y \gt 0}$;   solve for $y$

2. $\displaystyle{x^2 + y^2 = 9, \ \ y \le 0}$;   solve for $y$

3. $\displaystyle{x^2 + y^2 = 25, \ \ x \lt 0}$;   solve for $x$

4. $\displaystyle{x^2 + y^2 = 7, \ \ x \ge 0}$;   solve for $x$

5. $\displaystyle{x=y^2 + 1, \ \ x \lt 0}$;   solve for $y$

6. $\displaystyle{x=-2y^2, \ \ x \ge 0};$   solve for $y$

7. $\displaystyle{x^2 + y^2 = 0}$   solve for $y$

1. $y = \sqrt{16-x^2}$ with $x \neq \pm4$

2. $y = -\sqrt{9-x^2}$

3. $x = -\sqrt{25-x^2}$ with $y \neq \pm \sqrt{7}$

4. $x = \sqrt{7-y^2}$

5. no solution

6. $y = 0$

7. $y = 0$

3. Use an online graphing tool (like the one at www.desmos.com/calculator to graph the following relation $$\log_x y^2 + x = y$$ Based on what results, what appears to be the minimum number of functions needed to fully describe this relation? (Hint: recall that for a relation whose graph was a circle, we needed two functions)

$3$