Chapter 1 Section 6

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Take \[f(x)=\begin{cases} -x &\text{ if } -3< x \le 1\\ x^2-1 &\text{ if } 1< x \le 3 \\ x-7 &\text{ if } 3< x \le 7. \end{cases}\] Evaluate \(f\) at \(x=-3,\,x=0,\,x=1,\,x=3,\,x=5,\,x=8\).
  2. Take \[f(x)=\begin{cases} -x &\text{ if } -3< x \le 1\\ x^2-1 &\text{ if } 1< x \le 3 \\ x-7 &\text{ if } 3< x \le 7. \end{cases}\] Determine the domain of \(f.\) Use a sketch of the function \(f\) to determine the range of \(f\).
  3. Take \[A=\{(-3,0],(0,3],(3,7]\}\quad\text{and}\quad B=\{(-4,1],(1,2],(2,8]\}.\] Find a common refinement for \(A\) and \(B\).
  4. Take \[f(x)=\begin{cases} -x &\text{ if } -3< x \le 1\\ x^2-1 &\text{ if } 1< x \le 3 \\ x-7 &\text{ if } 3< x \le 7 \end{cases}\quad\text{and}\quad g(x)=\begin{cases} 2x-1 &\text{ if } -4< x \le 1\\ -3x+9 &\text{ if } 1< x \le 8. \end{cases}\] Determine \(\mathcal{D}(f+g)\), \(\mathcal{D}(fg)\), and \(\mathcal{D}\left(\frac{f}{g}\right).\)
  5. Take \[f(x)=\begin{cases} -x &\text{ if } -3< x \le 1\\ x^2-1 &\text{ if } 1< x \le 3 \\ x-7 &\text{ if } 3< x \le 7 \\ \end{cases}\quad\text{and}\quad g(x)=\begin{cases} 2x-1 &\text{ if } -4< x \le 1\\ -3x+9 &\text{ if } 1< x \le 8. \end{cases}\] Compute \(f+g\), \(fg\), and \(\frac{f}{g}.\)
  6. Take \[f(x)=\begin{cases} 2x-1 &\text{ if } x \le 1\\ -3x+9 &\text{ if } x>1. \end{cases}\] Sketch on a number line the solution set to the inequality \(f(x)>0\).
  7. Take \[f(x)=\begin{cases} 2x-1 &\text{ if } x \le 1\\ -3x+9 &\text{ if } x>1\\ \end{cases}\quad \text{and}\quad g(x)=\begin{cases} -1.5 &\text{ if } x \le -1\\ 3x-9 &\text{ if } x>-1. \end{cases}\] Sketch on a number line the solution set to the inequality \(f(x)>g(x)\).
  8. Sketch on a number line the solution set to the inequality \[|x-2|>|3x+1|.\]

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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