Question

(1 point) A function f is said to have a removable discontinuity at a if: 1. f is either not defined or not continuous at a. 2. f(a) could either be defined or redefined so that the new function is continuous at a. Let f(x) = 2x2+6x–80 X-5 Show that f has a removable discontinuity at 5 and determine the value for f(5) that would make f continuous at 5. Need to redefine f(5) =

Answer 1

f(x) = (2x² + 6x - 80) / (x - 5)

= 2(x² + 3x - 40) / (x - 5)

= 2 (x² + 8x - 5x - 40) / (x - 5)

= 2 (x(x + 8) - 5(x + 8)) / (x - 5)

= 2 (x - 5) (x + 8) / (x - 5)

Since the factor (x - 5) is common in the numerator and denominator of the function f(x), this implies there exists a hole or removable discontinuity at x = 5

f(x) = 2 (x - 5) (x + 8) / (x - 5)

=> f(x) = 2 (x + 8)

Therefore,

f(5) = 2 (5 + 8) = 26

Therefore,

In order to make f continuous at 5, Need to redefine f(5) = 26