Precalculus Examples

$f (x) = \frac{6}{\sqrt{8 - x}}$

Step 1

Write $f (x) = \frac{6}{\sqrt{8 - x}}$ as an equation.

$y = \frac{6}{\sqrt{8 - x}}$

Step 2

Interchange the variables.

$x = \frac{6}{\sqrt{8 - y}}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{6}{\sqrt{8 - y}} = x$ .

$\frac{6}{\sqrt{8 - y}} = x$

Step 3.2

Cross multiply.

Tap for more steps...

Step 3.2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$x \cdot (\sqrt{8 - y}) = 6$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Multiply $x$ by $\sqrt{8 - y}$ .

$x \sqrt{8 - y} = 6$

Step 3.3

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

${(x \sqrt{8 - y})}^{2} = 6^{2}$

Step 3.4

Simplify each side of the equation.

Tap for more steps...

Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - y}$ as ${(8 - y)}^{\frac{1}{2}}$ .

${(x {(8 - y)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify ${(x {(8 - y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.4.2.1.1

Apply the product rule to $x {(8 - y)}^{\frac{1}{2}}$ .

$x^{2} {({(8 - y)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.4.2.1.2

Multiply the exponents in ${({(8 - y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.4.2.1.2.1

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

$x^{2} {(8 - y)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.4.2.1.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.2.1.2.2.1

Cancel the common factor.

$x^{2} {(8 - y)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.4.2.1.2.2.2

Rewrite the expression.

$x^{2} {(8 - y)}^{1} = 6^{2}$

Step 3.4.2.1.3

Simplify.

$x^{2} (8 - y) = 6^{2}$

Step 3.4.2.1.4

Apply the distributive property.

$x^{2} \cdot 8 + x^{2} (- y) = 6^{2}$

Step 3.4.2.1.5

Reorder.

Tap for more steps...

Step 3.4.2.1.5.1

Move $8$ to the left of $x^{2}$ .

$8 \cdot x^{2} + x^{2} (- y) = 6^{2}$

Step 3.4.2.1.5.2

Rewrite using the commutative property of multiplication.

$8 x^{2} - x^{2} y = 6^{2}$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Raise $6$ to the power of $2$ .

$8 x^{2} - x^{2} y = 36$

Step 3.5

Solve for $y$ .

Tap for more steps...

Step 3.5.1

Subtract $8 x^{2}$ from both sides of the equation.

$- x^{2} y = 36 - 8 x^{2}$

Step 3.5.2

Divide each term in $- x^{2} y = 36 - 8 x^{2}$ by $- x^{2}$ and simplify.

Tap for more steps...

Step 3.5.2.1

Divide each term in $- x^{2} y = 36 - 8 x^{2}$ by $- x^{2}$ .

$\frac{- x^{2} y}{- x^{2}} = \frac{36}{- x^{2}} + \frac{- 8 x^{2}}{- x^{2}}$

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2} y}{x^{2}} = \frac{36}{- x^{2}} + \frac{- 8 x^{2}}{- x^{2}}$

Step 3.5.2.2.2

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.5.2.2.2.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{36}{- x^{2}} + \frac{- 8 x^{2}}{- x^{2}}$

Step 3.5.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{36}{- x^{2}} + \frac{- 8 x^{2}}{- x^{2}}$

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

Simplify each term.

Tap for more steps...

Step 3.5.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{36}{x^{2}} + \frac{- 8 x^{2}}{- x^{2}}$

Step 3.5.2.3.1.2

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.5.2.3.1.2.1

Cancel the common factor.

$y = - \frac{36}{x^{2}} + \frac{- 8 x^{2}}{- x^{2}}$

Step 3.5.2.3.1.2.2

Rewrite the expression.

$y = - \frac{36}{x^{2}} + \frac{- 8}{- 1}$

Step 3.5.2.3.1.2.3

Move the negative one from the denominator of $\frac{- 8}{- 1}$ .

$y = - \frac{36}{x^{2}} - 1 \cdot - 8$

Step 3.5.2.3.1.3

Rewrite $- 1 \cdot - 8$ as $- - 8$ .

$y = - \frac{36}{x^{2}} - - 8$

Step 3.5.2.3.1.4

Multiply $- 1$ by $- 8$ .

$y = - \frac{36}{x^{2}} + 8$

Step 4

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

$f^{- 1} (x) = - \frac{36}{x^{2}} + 8$

Step 5

Verify if $f^{- 1} (x) = - \frac{36}{x^{2}} + 8$ is the inverse of $f (x) = \frac{6}{\sqrt{8 - x}}$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{6}{\sqrt{8 - x}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{{(\frac{6}{\sqrt{8 - x}})}^{2}} + 8$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Simplify the denominator.

Tap for more steps...

Step 5.2.3.1.1

Apply the product rule to $\frac{6}{\sqrt{8 - x}}$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{6^{2}}{{\sqrt{8 - x}}^{2}}} + 8$

Step 5.2.3.1.2

Raise $6$ to the power of $2$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{{\sqrt{8 - x}}^{2}}} + 8$

Step 5.2.3.1.3

Rewrite ${\sqrt{8 - x}}^{2}$ as $8 - x$ .

Tap for more steps...

Step 5.2.3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - x}$ as ${(8 - x)}^{\frac{1}{2}}$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{{({(8 - x)}^{\frac{1}{2}})}^{2}}} + 8$

Step 5.2.3.1.3.2

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

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{{(8 - x)}^{\frac{1}{2} \cdot 2}}} + 8$

Step 5.2.3.1.3.3

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

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{{(8 - x)}^{\frac{2}{2}}}} + 8$

Step 5.2.3.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.3.1.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{{(8 - x)}^{\frac{2}{2}}}} + 8$

Step 5.2.3.1.3.4.2

Rewrite the expression.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{8 - x}} + 8$

Step 5.2.3.1.3.5

Simplify.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - \frac{36}{\frac{36}{8 - x}} + 8$

Step 5.2.3.2

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - (36 (\frac{8 - x}{36})) + 8$

Step 5.2.3.3

Cancel the common factor of $36$ .

Tap for more steps...

Step 5.2.3.3.1

Cancel the common factor.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - (36 (\frac{8 - x}{36})) + 8$

Step 5.2.3.3.2

Rewrite the expression.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - (8 - x) + 8$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - 1 \cdot 8 + x + 8$

Step 5.2.3.5

Multiply $- 1$ by $8$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - 8 + x + 8$

Step 5.2.3.6

Multiply $- - x$ .

Tap for more steps...

Step 5.2.3.6.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - 8 + 1 x + 8$

Step 5.2.3.6.2

Multiply $x$ by $1$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = - 8 + x + 8$

Step 5.2.4

Combine the opposite terms in $- 8 + x + 8$ .

Tap for more steps...

Step 5.2.4.1

Add $- 8$ and $8$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{6}{\sqrt{8 - x}}) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (- \frac{36}{x^{2}} + 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{8 - (- \frac{36}{x^{2}} + 8)}}$

Step 5.3.3

Simplify the denominator.

Tap for more steps...

Step 5.3.3.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{8 - (- \frac{36}{x^{2}} + \frac{8 x^{2}}{x^{2}})}}$

Step 5.3.3.2

Combine the numerators over the common denominator.

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{8 - \frac{- 36 + 8 x^{2}}{x^{2}}}}$

Step 5.3.3.3

Factor $4$ out of $- 36 + 8 x^{2}$ .

Tap for more steps...

Step 5.3.3.3.1

Factor $4$ out of $- 36$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{8 - \frac{4 (- 9) + 8 x^{2}}{x^{2}}}}$

Step 5.3.3.3.2

Factor $4$ out of $8 x^{2}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{8 - \frac{4 (- 9) + 4 (2 x^{2})}{x^{2}}}}$

Step 5.3.3.3.3

Factor $4$ out of $4 (- 9) + 4 (2 x^{2})$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{8 - \frac{4 (- 9 + 2 x^{2})}{x^{2}}}}$

Step 5.3.3.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{8 x^{2}}{x^{2}} - \frac{4 (- 9 + 2 x^{2})}{x^{2}}}}$

Step 5.3.3.5

Combine the numerators over the common denominator.

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{8 x^{2} - 4 (- 9 + 2 x^{2})}{x^{2}}}}$

Step 5.3.3.6

Rewrite $\frac{8 x^{2} - 4 (- 9 + 2 x^{2})}{x^{2}}$ in a factored form.

Tap for more steps...

Step 5.3.3.6.1

Factor $4$ out of $8 x^{2} - 4 (- 9 + 2 x^{2})$ .

Tap for more steps...

Step 5.3.3.6.1.1

Factor $4$ out of $8 x^{2}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (2 x^{2}) - 4 (- 9 + 2 x^{2})}{x^{2}}}}$

Step 5.3.3.6.1.2

Factor $4$ out of $- 4 (- 9 + 2 x^{2})$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (2 x^{2}) + 4 (- (- 9 + 2 x^{2}))}{x^{2}}}}$

Step 5.3.3.6.1.3

Factor $4$ out of $4 (2 x^{2}) + 4 (- (- 9 + 2 x^{2}))$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (2 x^{2} - (- 9 + 2 x^{2}))}{x^{2}}}}$

Step 5.3.3.6.2

Apply the distributive property.

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (2 x^{2} + 9 - (2 x^{2}))}{x^{2}}}}$

Step 5.3.3.6.3

Multiply $- 1$ by $- 9$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (2 x^{2} + 9 - (2 x^{2}))}{x^{2}}}}$

Step 5.3.3.6.4

Multiply $2$ by $- 1$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (2 x^{2} + 9 - 2 x^{2})}{x^{2}}}}$

Step 5.3.3.6.5

Subtract $2 x^{2}$ from $2 x^{2}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 (0 + 9)}{x^{2}}}}$

Step 5.3.3.6.6

Add $0$ and $9$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{4 \cdot 9}{x^{2}}}}$

Step 5.3.3.7

Multiply $4$ by $9$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{36}{x^{2}}}}$

Step 5.3.3.8

Rewrite $36$ as $6^{2}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{\frac{6^{2}}{x^{2}}}}$

Step 5.3.3.9

Rewrite $\frac{6^{2}}{x^{2}}$ as ${(\frac{6}{x})}^{2}$ .

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\sqrt{{(\frac{6}{x})}^{2}}}$

Step 5.3.3.10

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

$f (- \frac{36}{x^{2}} + 8) = \frac{6}{\frac{6}{x}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{36}{x^{2}} + 8) = 6 (\frac{x}{6})$

Step 5.3.5

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.3.5.1

Cancel the common factor.

$f (- \frac{36}{x^{2}} + 8) = 6 (\frac{x}{6})$

Step 5.3.5.2

Rewrite the expression.

$f (- \frac{36}{x^{2}} + 8) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{36}{x^{2}} + 8$ is the inverse of $f (x) = \frac{6}{\sqrt{8 - x}}$ .

$f^{- 1} (x) = - \frac{36}{x^{2}} + 8$