Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Divide each term in $- \sqrt{y + 1} = x$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \sqrt{y + 1} = x$ by $- 1$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

Divide $\sqrt{y + 1}$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

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

Step 3.2.3.2

Rewrite $- 1 \cdot x$ as $- x$ .

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

Step 3.3

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

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

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

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

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(- x)}^{2}$ .

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Step 3.4.3.1.1

Apply the product rule to $- x$ .

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

Step 3.4.3.1.2

Raise $- 1$ to the power of $2$ .

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

Step 3.4.3.1.3

Multiply $x^{2}$ by $1$ .

$y + 1 = x^{2}$

Step 3.5

Subtract $1$ from both sides of the equation.

$y = x^{2} - 1$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 5

Verify if $f^{- 1} (x) = x^{2} - 1$ is the inverse of $f (x) = - \sqrt{x + 1}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (- \sqrt{x + 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the product rule to $- \sqrt{x + 1}$ .

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

Step 5.2.3.2

Raise $- 1$ to the power of $2$ .

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

Step 5.2.3.3

Multiply ${\sqrt{x + 1}}^{2}$ by $1$ .

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

Step 5.2.3.4

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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Step 5.2.3.4.1

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

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

Step 5.2.3.4.2

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

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

Step 5.2.3.4.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.2.3.4.4

Cancel the common factor of $2$ .

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Step 5.2.3.4.4.1

Cancel the common factor.

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

Step 5.2.3.4.4.2

Rewrite the expression.

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

Step 5.2.3.4.5

Simplify.

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

Step 5.2.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 5.2.4.1

Subtract $1$ from $1$ .

$f^{- 1} (- \sqrt{x + 1}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (x^{2} - 1)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Add $- 1$ and $1$ .

$f (x^{2} - 1) = - \sqrt{x^{2} + 0}$

Step 5.3.4

Add $x^{2}$ and $0$ .

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

Step 5.3.5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} - 1$ is the inverse of $f (x) = - \sqrt{x + 1}$ .

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