Question

An expression is well-defined if you can compute its value without any illegal operations. examples of expressions that are not well-defined include $1/0$ and $\sqrt{-10}$. for what values of $x$ is the expression \[\frac{\sqrt{x + 1} + \sqrt{1 - x}}{\sqrt{x}}\] well-defined? express your answer in interval notation.

Answer 1

The given expression is
`(√(x + 1) + √(1 - x))/(√(x))`

So, we have 3 square roots.
The number under a square root sign must be Non negative.
Beside that the denominator must be not equal zero.
For the term ⇒⇒⇒
`√(x)`
∴ x > 0 ⇒⇒⇒ x ∈ (0,∞)
For the term ⇒⇒⇒
`√(x+1)`
∴x+1 ≥ 0 ⇒⇒⇒ x ≥ -1 ⇒⇒⇒ ∴ x ∈ [-1,∞)
For the term ⇒⇒⇒
`√(1-x)`
∴ 1-x ≥ 0 ⇒⇒⇒ x ≤ 1 ⇒⇒⇒ x ∈ (-∞,1]

So, The values of x which make the expression well defined

∴ x ∈ { (0,∞) ∪ [-1,∞) ∪ (-∞,1] }

∴ x ∈ [-1,0) ∪ (0,1]

OR can be written as ⇒⇒ x ∈ [-1,1] - {0}