How can I solve y=1/sqrt(1-x) — step-by-step solution

The expression

y = \frac{1}{\sqrt{1 - x}}

contains several variables: the variable x is chosen for solving

1) Let's solve in

ℝ

y = \frac{1}{\sqrt{1 - x}}

\frac{1}{\sqrt{1 - x}} = y

We multiply both sides by

\sqrt{1 - x}

, nonzero for the fractions to be defined

\frac{1}{\sqrt{1 - x}} \cdot \sqrt{1 - x} = y \cdot \sqrt{1 - x}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{\sqrt{1 - x}} \cdot \sqrt{1 - x}

\frac{1 \cdot \sqrt{1 - x}}{\sqrt{1 - x}}

\frac{\sqrt{1 - x}}{\sqrt{1 - x}}

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

\sqrt{1 - x}

\frac{\sqrt{1 - x}}{\sqrt{1 - x}}

1

1 = y \cdot \sqrt{1 - x}

y \cdot \sqrt{1 - x} = 1

Isolate the square root:

\sqrt{1 - x} = \frac{1}{y}

. Since the square root is nonnegative, square both sides, provided that

\frac{1}{y} \geq 0

1 - x = {(\frac{1}{y})}^{2}

ⓘ We gather the terms in

x

on the right and the other terms on the left

1 - {(\frac{1}{y})}^{2} = x

ⓘ The power of a quotient is the quotient of the powers, (a/b)^n = a^n/b^n:

{(\frac{1}{y})}^{2}

\frac{1}{y^{2}}

1 - \frac{1}{y^{2}} = x

Candidates, pending the domain check:

x = 1 - \frac{1}{y^{2}}

2) Solving for y:

\frac{1}{y} \geq 0

⇥The expression

y

is zero at

y = 0

(forbidden value), it is negative before and positive after

3) Let's draw up the sign table:

Candidates, pending the domain check:

y > 0

4) Let's find the domain of

y = \frac{1}{\sqrt{1 - x}}

For the expression

y = \frac{1}{\sqrt{1 - x}}

to be defined, we must have:

1 - x > 0

ⓘ We add

x

to both sides

1 > x

Domain constraints:

x < 1

▶Solution:

{x = 1 - \frac{1}{y^{2}}, y > 0}