Question

The function f(x) = x – |x – x²| is ______.

Options

• continuous at x = 1

• discontinuous at x = 1

• not defined at x = 1

• None of these

Answer 1

The function f(x) = x – |x – x²| is continuous at x = 1.

Explanation:

Given, f(x) = x – |x – x²|

At x = 1, f(1) = 1 – |1 – 1| = 1

`lim_(x rightarrow 1^-) f(x) = lim_(h rightarrow 0) [(1 - h) - |(1 - h) - (1 - h)^2|]`

= `lim_(h rightarrow 0) [(1 - h) - |h - h^2|]` = 1

`lim_(x rightarrow 1^+) f(x) = lim_(h rightarrow 0) [(1 + h) - |(1 + h) - (1 + h)^2|]`

= `lim_(h rightarrow 0) [1 + h - |-h^2 - h|]` = 1

∵ `lim_(x rightarrow 1^-) f(x) = lim_(x rightarrow 1^+)` = f(1)

∴ f(x) is continuous at x = 1.