Answers


  1. Answers may vary.

    1. Graph of y=x-4,x<3; y=3x-8,x>3

    2. Graph of y=0.5x+1/2,x<1; y=0.5x-2.5,x>1; point (1, 2)

    3. Graph of y=-3,x<-1; y=x-2,x>-1; point (-1, 3)
  2. Answers may vary.
    Graph of constant function y=2 with discontinuity at (-1,2).
    1. \( f \) is continuous at \( x = A \).
    2. \( f \) is discontinuous at \( x = B \) and it is a removable discontinuity.
    3. \( f \) is discontinuous at \( x = C \) and it is a jump discontinuity.
    4. \( f \) is continuous at \( x = D \).
    5. \( f \) is discontinuous at \( x = E \) and it is an infinite discontinuity.
    1. Infinite discontinuity at \( x = -1 \)
      Removable discontinuity at \(x=-2\)
    2. Jump discontinuity at \( x = 3 \)
    3. Infinite discontinuity at \(x=2\)
    1. all \( x \)
    2. all \( x \neq 0, 5 \)
    3. all \( x \)
    4. Continuous for \( x \gt -\frac{3}{2} \) and continuous from the right at \( x = -\frac{3}{2} \)
    1. Not continuous.
      Graph of piecewise function described in question; hole at (0,-3), jump discontinuity at x=0
    2. Continuous for all \( x \).
      Graph of piecewise function described in question
  7. Jump discontinuity.
    1. \( a = 2 \)
    2. \( a = 1, b = \frac{5}{3} \) or \( a = -1, b = 1 \)
  9. Answers may vary.

    1. Graph of y=-1,x<1;y=1,x>1;point (1,0)

    2. Graph of y=0,x<-1;y=-1,-1<x<3;y=x-2,x>3
  10. Answers may vary.

    1. Graph of y=x with a hole at (0,0)

    2. Graph of function with endpoint at (0,1), increasing on (0,1), function approaches infinity as x approaches 1

    3. Graph of y=1/(x-0.5) + 1 with a hole at (-1,1/3)

    4. Graph of y=3/(x - 1) for x<3, y=x for x>3, hole at (3,-1.5)
  11. Answers may vary.
    1. \( f(x) = \dfrac{x^{2}-2x}{x-2}, ~~ g(x) = \begin{cases} x^{2} &\text{if } x \neq 2 \\ 1 &\text{if } x=2 \end{cases} \)
    2. \(f(x) = \sqrt{x} + \dfrac{1}{\sqrt{1-x}}, ~~ g(x) = \begin{cases} -2 &\text{if } 0 \leq x \lt \frac{1}{2} \\ \dfrac{1}{x-1} &\text{if } \frac{1}{2} \leq x \lt 1 \\ 2 &\text{if } x =1 \end{cases} \)
    3. \( f(x) = \dfrac{x^{2}+x}{2x^{2} + x -1}\)
    4. \(f(x) = \begin{cases} \dfrac{3}{1-x} &\text{if } x \lt 3, x \neq 1 \\ x &\text{if } x \leq 3 \end{cases}\)