Discuss the continuity of the following functions at the point(s). f(x) = 2x^2 + x + 1, for |x – 3| ≥ 2 = x^2 + 3, for 1 < x < 5

Question

Discuss the continuity of the following functions at the point(s) or on the interval indicated against them.

f(x) = 2x² + x + 1, for |x – 3| ≥ 2 

= x² + 3, for 1 < x < 5

Answer 1

|x – 3| ≥ 2 

∴ x – 3 ≥ 2 or x – 3 ≤ -2 

∴ x ≥ 5 or x ≤ 1 

∴ f(x) = 2x² + x + 1, x ≤ 1 

= x² + 3, 1 < x < 5 

= 2x² + x + 1, x ≥ 5 

Consider the intervals 

x < 1 , i.e., (-∞, 1) 

1 < x < 5, i.e., (1, 5) x > 5, i.e., (5, ∞) 

In all these intervals, f(x) is a polynomial function and hence is continuous at all points. 

For continuity at x = 1:

∴ f(x) is discontinuous at x = 5. 

∴ f(x) is continuous for all x ∈ R, except at x = 5.