practice #2 wednesday\nmultiple choice: select the best answer for each question.\n1) the graph of the…

practice #2 wednesday\nmultiple choice: select the best answer for each question.\n1) the graph of the function (f(x)) is shown at right. which of the following statements is true about (f)? i. (f) is undefined at (x = 1). ii. (f) is defined but not continuous at (x = 2). iii. (f) is defined and continuous at (x = 3). a) only i b) only ii c) i and ii d) i and iii e) none of the statements are true.\nquestions 2 through 4 refer to the function (g(x)) shown at right.\n2) the function (g(x)) has a removable discontinuity at: a) (x=-2) only b) (x = 0) only c) (x = 1) only d) (x=-2) and (x = 0) only e) (g(x)) has no removable discontinuities.\n3) on what intervals is (g(x)) continuous? a) (-3,-2cup-2,0cup0,2.5) b) (-3,-2cup(-2,0cup0,2.5) c) (-3,-2cup(-2,0cup(0,2.5) d) (-3,-2cup-2,0cup(0,2.5) e) (-3,-2cup(-2,0cup(0,1)cup(1,2.5)

practice #2 wednesday\nmultiple choice: select the best answer for each question.\n1) the graph of the function (f(x)) is shown at right. which of the following statements is true about (f)? i. (f) is undefined at (x = 1). ii. (f) is defined but not continuous at (x = 2). iii. (f) is defined and continuous at (x = 3). a) only i b) only ii c) i and ii d) i and iii e) none of the statements are true.\nquestions 2 through 4 refer to the function (g(x)) shown at right.\n2) the function (g(x)) has a removable discontinuity at: a) (x=-2) only b) (x = 0) only c) (x = 1) only d) (x=-2) and (x = 0) only e) (g(x)) has no removable discontinuities.\n3) on what intervals is (g(x)) continuous? a) (-3,-2cup-2,0cup0,2.5) b) (-3,-2cup(-2,0cup0,2.5) c) (-3,-2cup(-2,0cup(0,2.5) d) (-3,-2cup-2,0cup(0,2.5) e) (-3,-2cup(-2,0cup(0,1)cup(1,2.5)

Answer

Explanation:

Step1: Analyze function (f(x)) at (x = 1)

The graph of (f(x)) has an open - circle at (x = 1), so (f) is undefined at (x=1).

Step2: Analyze function (f(x)) at (x = 2)

The graph of (f(x)) has a break at (x = 2). The function is defined (there is a closed - circle) but not continuous at (x = 2).

Step3: Analyze function (f(x)) at (x = 3)

The graph of (f(x)) has a vertical asymptote at (x = 3), so (f) is not continuous at (x = 3). Statements I and II are true.

Step4: Analyze function (g(x)) for removable discontinuity

A removable discontinuity occurs when there is a hole in the graph. For (g(x)), there is a hole at (x=-2), so it has a removable discontinuity at (x = - 2).

Step5: Determine continuity intervals of (g(x))

The function (g(x)) is continuous on ([-3,-2]\cup(-2,0]\cup(0,2.5]) because it is continuous within each of these sub - intervals and has non - removable discontinuities at the endpoints of the unioned intervals as appropriate.

Answer:

  1. C) I and II
  2. A) (x=-2) only
  3. C) ([-3,-2]\cup(-2,0]\cup(0,2.5])