Precalculus Examples

$f (x) = \sqrt{1 - x^{2}}$

Step 1

Write $f (x) = \sqrt{1 - x^{2}}$ as an equation.

$y = \sqrt{1 - x^{2}}$

Step 2

Interchange the variables.

$x = \sqrt{1 - y^{2}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{1 - y^{2}} = x$ .

$\sqrt{1 - y^{2}} = x$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{1 - y^{2}}}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - y^{2}}$ as ${(1 - y^{2})}^{\frac{1}{2}}$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(1 - y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(1 - y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

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

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(1 - y^{2})}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$1 - y^{2} = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $1$ from both sides of the equation.

$- y^{2} = x^{2} - 1$

Step 3.4.2

Divide each term in $- y^{2} = x^{2} - 1$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- y^{2} = x^{2} - 1$ by $- 1$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.4.2.2.2

Divide $y^{2}$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

$y^{2} = - 1 \cdot x^{2} + \frac{- 1}{- 1}$

Step 3.4.2.3.1.2

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

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

Step 3.4.2.3.1.3

Divide $- 1$ by $- 1$ .

$y^{2} = - x^{2} + 1$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- x^{2} + 1}$

Step 3.4.4

Simplify $\pm \sqrt{- x^{2} + 1}$ .

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Step 3.4.4.1

Simplify the expression.

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Step 3.4.4.1.1

Rewrite $1$ as $1^{2}$ .

$y = \pm \sqrt{- x^{2} + 1^{2}}$

Step 3.4.4.1.2

Reorder $- x^{2}$ and $1^{2}$ .

$y = \pm \sqrt{1^{2} - x^{2}}$

Step 3.4.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \pm \sqrt{(1 + x) (1 - x)}$

Step 3.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{(1 + x) (1 - x)}$

Step 3.4.5.2

Next, use the negative value of the $\pm$ to find the second solution.