Question

Which of the following functions are discontinuous?

Answer 1

Answer:

D.  I, II, and III

Step-by-step explanation:

A discontinuous function is a function which is not continuous.  

If f(x) is not continuous at x = a, then f(x) is said to be discontinuous at this point.

To prove whether a function is discontinuous, find where it is undefined.

A rational function is undefined when the denominator is equal to zero.

Therefore, to find the values that make a rational function undefined, set the denominator to zero and solve.

Function I

Denominator:  x - 2

Set to zero:  x - 2 = 0

Solve:  x = 2

Therefore, this function is undefined when x = 2 and so the function is discontinuous.

Function II

Denominator:  4x²

Set to zero:  4x² = 0

Solve:  x = 0

Therefore, this function is undefined when x = 0 and so the function is discontinuous.

Function III

Denominator:  x² + 3x + 2

Set to zero:  x² + 3x + 2 = 0

Solve:  

⇒ x² + 3x + 2 = 0

⇒ x² + x + 2x + 2 = 0

⇒ x(x + 1) + 2(x + 1) = 0

⇒ (x + 2)(x + 1) = 0

⇒ x = -2, x = -1

Therefore, this function is undefined when x = -2 and x = -1, and so the function is discontinuous.

Therefore, all three given functions are discontinuous.