Finite Math Examples

$f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$

Step 1

Write $f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$ as an equation.

$y = \frac{\sqrt{2 x + 3}}{x - 8}$

Step 2

Interchange the variables.

$x = \frac{\sqrt{2 y + 3}}{y - 8}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 8$ .

$(y - 8) (x) = (\frac{\sqrt{2 y + 3}}{y - 8}) (y - 8)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y x - 8 x = (\frac{\sqrt{2 y + 3}}{y - 8}) (y - 8)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y - 8$ .

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Step 3.3.1.1

Cancel the common factor.

$y x - 8 x = \frac{\sqrt{2 y + 3}}{y - 8} (y - 8)$

Step 3.3.1.2

Rewrite the expression.

$y x - 8 x = \sqrt{2 y + 3}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Rewrite the equation as $\sqrt{2 y + 3} = y x - 8 x$ .

$\sqrt{2 y + 3} = y x - 8 x$

Step 3.4.2

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

${\sqrt{2 y + 3}}^{2} = {(y x - 8 x)}^{2}$

Step 3.4.3

Simplify each side of the equation.

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Step 3.4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 y + 3}$ as ${(2 y + 3)}^{\frac{1}{2}}$ .

${({(2 y + 3)}^{\frac{1}{2}})}^{2} = {(y x - 8 x)}^{2}$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Simplify ${({(2 y + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.3.2.1.1

Multiply the exponents in ${({(2 y + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.3.2.1.1.1

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

${(2 y + 3)}^{\frac{1}{2} \cdot 2} = {(y x - 8 x)}^{2}$

Step 3.4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.3.2.1.1.2.1

Cancel the common factor.

${(2 y + 3)}^{\frac{1}{2} \cdot 2} = {(y x - 8 x)}^{2}$

Step 3.4.3.2.1.1.2.2

Rewrite the expression.

${(2 y + 3)}^{1} = {(y x - 8 x)}^{2}$

Step 3.4.3.2.1.2

Simplify.

$2 y + 3 = {(y x - 8 x)}^{2}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

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

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Step 3.4.3.3.1.1

Rewrite ${(y x - 8 x)}^{2}$ as $(y x - 8 x) (y x - 8 x)$ .

$2 y + 3 = (y x - 8 x) (y x - 8 x)$

Step 3.4.3.3.1.2

Expand $(y x - 8 x) (y x - 8 x)$ using the FOIL Method.

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Step 3.4.3.3.1.2.1

Apply the distributive property.

$2 y + 3 = y x (y x - 8 x) - 8 x (y x - 8 x)$

Step 3.4.3.3.1.2.2

Apply the distributive property.

$2 y + 3 = y x (y x) + y x (- 8 x) - 8 x (y x - 8 x)$

Step 3.4.3.3.1.2.3

Apply the distributive property.

$2 y + 3 = y x (y x) + y x (- 8 x) - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3

Simplify and combine like terms.

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Step 3.4.3.3.1.3.1

Simplify each term.

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Step 3.4.3.3.1.3.1.1

Multiply $y$ by $y$ by adding the exponents.

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Step 3.4.3.3.1.3.1.1.1

Move $y$ .

$2 y + 3 = y \cdot y x \cdot x + y x (- 8 x) - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.1.2

Multiply $y$ by $y$ .

$2 y + 3 = y^{2} x \cdot x + y x (- 8 x) - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.3.3.1.3.1.2.1

Move $x$ .

$2 y + 3 = y^{2} (x \cdot x) + y x (- 8 x) - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$2 y + 3 = y^{2} x^{2} + y x (- 8 x) - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$2 y + 3 = y^{2} x^{2} - 8 y x \cdot x - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.3.3.1.3.1.4.1

Move $x$ .

$2 y + 3 = y^{2} x^{2} - 8 y (x \cdot x) - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.4.2

Multiply $x$ by $x$ .

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 x (y x) - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.3.3.1.3.1.5.1

Move $x$ .

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 (x \cdot x) y - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 x^{2} y - 8 x (- 8 x)$

Step 3.4.3.3.1.3.1.6

Rewrite using the commutative property of multiplication.

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 x^{2} y - 8 \cdot - 8 x \cdot x$

Step 3.4.3.3.1.3.1.7

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.3.3.1.3.1.7.1

Move $x$ .

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 x^{2} y - 8 \cdot - 8 (x \cdot x)$

Step 3.4.3.3.1.3.1.7.2

Multiply $x$ by $x$ .

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 x^{2} y - 8 \cdot - 8 x^{2}$

Step 3.4.3.3.1.3.1.8

Multiply $- 8$ by $- 8$ .

$2 y + 3 = y^{2} x^{2} - 8 y x^{2} - 8 x^{2} y + 64 x^{2}$

Step 3.4.3.3.1.3.2

Subtract $8 x^{2} y$ from $- 8 y x^{2}$ .

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Step 3.4.3.3.1.3.2.1

Move $y$ .

$2 y + 3 = y^{2} x^{2} - 8 x^{2} y - 8 x^{2} y + 64 x^{2}$

Step 3.4.3.3.1.3.2.2

Subtract $8 x^{2} y$ from $- 8 x^{2} y$ .

$2 y + 3 = y^{2} x^{2} - 16 x^{2} y + 64 x^{2}$

Step 3.4.4

Solve for $y$ .

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Step 3.4.4.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y^{2} x^{2} - 16 x^{2} y + 64 x^{2} = 2 y + 3$

Step 3.4.4.2

Subtract $2 y$ from both sides of the equation.

$y^{2} x^{2} - 16 x^{2} y + 64 x^{2} - 2 y = 3$

Step 3.4.4.3

Subtract $3$ from both sides of the equation.

$y^{2} x^{2} - 16 x^{2} y + 64 x^{2} - 2 y - 3 = 0$

Step 3.4.4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4.4.5

Substitute the values $a = x^{2}$ , $b = - 16 x^{2} - 2$ , and $c = 64 x^{2} - 3$ into the quadratic formula and solve for $y$ .

$\frac{- (- 16 x^{2} - 2) \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot (x^{2} \cdot (64 x^{2} - 3))}}{2 x^{2}}$

Step 3.4.4.6

Simplify the numerator.

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Step 3.4.4.6.1

Apply the distributive property.

$y = \frac{- (- 16 x^{2}) + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot x^{2} \cdot (64 x^{2} - 3)}}{2 x^{2}}$

Step 3.4.4.6.2

Multiply $- 16$ by $- 1$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot x^{2} \cdot (64 x^{2} - 3)}}{2 x^{2}}$

Step 3.4.4.6.3

Multiply $- 1$ by $- 2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot x^{2} \cdot (64 x^{2} - 3)}}{2 x^{2}}$

Step 3.4.4.6.4

Add parentheses.

$y = \frac{16 x^{2} + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot (x^{2} \cdot (64 x^{2} - 3))}}{2 x^{2}}$

Step 3.4.4.6.5

Let $u = x^{2} \cdot (64 x^{2} - 3)$ . Substitute $u$ for all occurrences of $x^{2} \cdot (64 x^{2} - 3)$ .

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Step 3.4.4.6.5.1

Rewrite ${(- 16 x^{2} - 2)}^{2}$ as $(- 16 x^{2} - 2) (- 16 x^{2} - 2)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{(- 16 x^{2} - 2) (- 16 x^{2} - 2) - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.2

Expand $(- 16 x^{2} - 2) (- 16 x^{2} - 2)$ using the FOIL Method.

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Step 3.4.4.6.5.2.1

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 x^{2} (- 16 x^{2} - 2) - 2 (- 16 x^{2} - 2) - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.2.2

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 x^{2} (- 16 x^{2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2} - 2) - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.2.3

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 x^{2} (- 16 x^{2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3

Simplify and combine like terms.

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Step 3.4.4.6.5.3.1

Simplify each term.

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Step 3.4.4.6.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 x^{2} x^{2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.4.4.6.5.3.1.2.1

Move $x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 (x^{2} x^{2})) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 x^{2 + 2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.2.3

Add $2$ and $2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 x^{4}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.3

Multiply $- 16$ by $- 16$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.4

Multiply $- 2$ by $- 16$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 32 x^{2} - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.5

Multiply $- 16$ by $- 2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 32 x^{2} + 32 x^{2} - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.1.6

Multiply $- 2$ by $- 2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 32 x^{2} + 32 x^{2} + 4 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.6.5.3.2

Add $32 x^{2}$ and $32 x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 64 x^{2} + 4 - 4 \cdot u}}{2 x^{2}}$

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 64 x^{2} + 4 - 4 u}}{2 x^{2}}$

Step 3.4.4.6.6

Factor $4$ out of $256 x^{4} + 64 x^{2} + 4 - 4 u$ .

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Step 3.4.4.6.6.1

Factor $4$ out of $256 x^{4}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 64 x^{2} + 4 - 4 u}}{2 x^{2}}$

Step 3.4.4.6.6.2

Factor $4$ out of $64 x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 4 (16 x^{2}) + 4 - 4 u}}{2 x^{2}}$

Step 3.4.4.6.6.3

Factor $4$ out of $4$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 4 (16 x^{2}) + 4 (1) - 4 u}}{2 x^{2}}$

Step 3.4.4.6.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 4 (16 x^{2}) + 4 (1) + 4 (- u)}}{2 x^{2}}$

Step 3.4.4.6.6.5

Factor $4$ out of $4 (64 x^{4}) + 4 (16 x^{2})$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2}) + 4 (1) + 4 (- u)}}{2 x^{2}}$

Step 3.4.4.6.6.6

Factor $4$ out of $4 (64 x^{4} + 16 x^{2}) + 4 (1)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1) + 4 (- u)}}{2 x^{2}}$

Step 3.4.4.6.6.7

Factor $4$ out of $4 (64 x^{4} + 16 x^{2} + 1) + 4 (- u)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - u)}}{2 x^{2}}$

Step 3.4.4.6.7

Replace all occurrences of $u$ with $x^{2} \cdot (64 x^{2} - 3)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (x^{2} \cdot (64 x^{2} - 3)))}}{2 x^{2}}$

Step 3.4.4.6.8

Simplify.

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Step 3.4.4.6.8.1

Simplify each term.

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Step 3.4.4.6.8.1.1

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (x^{2} (64 x^{2}) + x^{2} \cdot - 3))}}{2 x^{2}}$

Step 3.4.4.6.8.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{2} x^{2} + x^{2} \cdot - 3))}}{2 x^{2}}$

Step 3.4.4.6.8.1.3

Move $- 3$ to the left of $x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{2} x^{2} - 3 \cdot x^{2}))}}{2 x^{2}}$

Step 3.4.4.6.8.1.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.4.4.6.8.1.4.1

Move $x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 (x^{2} x^{2}) - 3 \cdot x^{2}))}}{2 x^{2}}$

Step 3.4.4.6.8.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{2 + 2} - 3 \cdot x^{2}))}}{2 x^{2}}$

Step 3.4.4.6.8.1.4.3

Add $2$ and $2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{4} - 3 \cdot x^{2}))}}{2 x^{2}}$

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{4} - 3 x^{2}))}}{2 x^{2}}$

Step 3.4.4.6.8.1.5

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{4}) - (- 3 x^{2}))}}{2 x^{2}}$

Step 3.4.4.6.8.1.6

Multiply $64$ by $- 1$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - 64 x^{4} - (- 3 x^{2}))}}{2 x^{2}}$

Step 3.4.4.6.8.1.7

Multiply $- 3$ by $- 1$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - 64 x^{4} + 3 x^{2})}}{2 x^{2}}$

Step 3.4.4.6.8.2

Combine the opposite terms in $64 x^{4} + 16 x^{2} + 1 - 64 x^{4} + 3 x^{2}$ .

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Step 3.4.4.6.8.2.1

Subtract $64 x^{4}$ from $64 x^{4}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (16 x^{2} + 1 + 0 + 3 x^{2})}}{2 x^{2}}$

Step 3.4.4.6.8.2.2

Add $16 x^{2} + 1$ and $0$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (16 x^{2} + 1 + 3 x^{2})}}{2 x^{2}}$

Step 3.4.4.6.8.3

Add $16 x^{2}$ and $3 x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (19 x^{2} + 1)}}{2 x^{2}}$

Step 3.4.4.6.9

Rewrite $4 (19 x^{2} + 1)$ as $2^{2} (19 x^{2} + 1^{2})$ .

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Step 3.4.4.6.9.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{2^{2} (19 x^{2} + 1)}}{2 x^{2}}$

Step 3.4.4.6.9.2

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

$y = \frac{16 x^{2} + 2 \pm \sqrt{2^{2} (19 x^{2} + 1^{2})}}{2 x^{2}}$

Step 3.4.4.6.10

Pull terms out from under the radical.

$y = \frac{16 x^{2} + 2 \pm 2 \sqrt{19 x^{2} + 1^{2}}}{2 x^{2}}$

Step 3.4.4.6.11

One to any power is one.

$y = \frac{16 x^{2} + 2 \pm 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.4.4.7.1

Change the $\pm$ to $+$ .

$y = \frac{16 x^{2} + 2 + 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.7.2

Cancel the common factor of $16 x^{2} + 2 + 2 \sqrt{19 x^{2} + 1}$ and $2$ .

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Step 3.4.4.7.2.1

Factor $2$ out of $16 x^{2}$ .

$y = \frac{2 (8 x^{2}) + 2 + 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.7.2.2

Factor $2$ out of $2$ .

$y = \frac{2 (8 x^{2}) + 2 \cdot 1 + 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.7.2.3

Factor $2$ out of $2 (8 x^{2}) + 2 (1)$ .

$y = \frac{2 (8 x^{2} + 1) + 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.7.2.4

Factor $2$ out of $2 \sqrt{19 x^{2} + 1}$ .

$y = \frac{2 (8 x^{2} + 1) + 2 (\sqrt{19 x^{2} + 1})}{2 x^{2}}$

Step 3.4.4.7.2.5

Factor $2$ out of $2 (8 x^{2} + 1) + 2 (\sqrt{19 x^{2} + 1})$ .

$y = \frac{2 (8 x^{2} + 1 + \sqrt{19 x^{2} + 1})}{2 x^{2}}$

Step 3.4.4.7.2.6

Cancel the common factors.

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Step 3.4.4.7.2.6.1

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

$y = \frac{2 (8 x^{2} + 1 + \sqrt{19 x^{2} + 1})}{2 (x^{2})}$

Step 3.4.4.7.2.6.2

Cancel the common factor.

$y = \frac{2 (8 x^{2} + 1 + \sqrt{19 x^{2} + 1})}{2 x^{2}}$

Step 3.4.4.7.2.6.3

Rewrite the expression.

$y = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$

Step 3.4.4.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.4.4.8.1

Simplify the numerator.

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Step 3.4.4.8.1.1

Apply the distributive property.

$y = \frac{- (- 16 x^{2}) + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot x^{2} \cdot (64 x^{2} - 3)}}{2 x^{2}}$

Step 3.4.4.8.1.2

Multiply $- 16$ by $- 1$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot x^{2} \cdot (64 x^{2} - 3)}}{2 x^{2}}$

Step 3.4.4.8.1.3

Multiply $- 1$ by $- 2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot x^{2} \cdot (64 x^{2} - 3)}}{2 x^{2}}$

Step 3.4.4.8.1.4

Add parentheses.

$y = \frac{16 x^{2} + 2 \pm \sqrt{{(- 16 x^{2} - 2)}^{2} - 4 \cdot (x^{2} \cdot (64 x^{2} - 3))}}{2 x^{2}}$

Step 3.4.4.8.1.5

Let $u = x^{2} \cdot (64 x^{2} - 3)$ . Substitute $u$ for all occurrences of $x^{2} \cdot (64 x^{2} - 3)$ .

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Step 3.4.4.8.1.5.1

Rewrite ${(- 16 x^{2} - 2)}^{2}$ as $(- 16 x^{2} - 2) (- 16 x^{2} - 2)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{(- 16 x^{2} - 2) (- 16 x^{2} - 2) - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.2

Expand $(- 16 x^{2} - 2) (- 16 x^{2} - 2)$ using the FOIL Method.

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Step 3.4.4.8.1.5.2.1

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 x^{2} (- 16 x^{2} - 2) - 2 (- 16 x^{2} - 2) - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.2.2

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 x^{2} (- 16 x^{2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2} - 2) - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.2.3

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 x^{2} (- 16 x^{2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3

Simplify and combine like terms.

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Step 3.4.4.8.1.5.3.1

Simplify each term.

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Step 3.4.4.8.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 x^{2} x^{2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.4.4.8.1.5.3.1.2.1

Move $x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 (x^{2} x^{2})) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 x^{2 + 2}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.2.3

Add $2$ and $2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{- 16 \cdot (- 16 x^{4}) - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.3

Multiply $- 16$ by $- 16$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} - 16 x^{2} \cdot - 2 - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.4

Multiply $- 2$ by $- 16$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 32 x^{2} - 2 (- 16 x^{2}) - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.5

Multiply $- 16$ by $- 2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 32 x^{2} + 32 x^{2} - 2 \cdot - 2 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.1.6

Multiply $- 2$ by $- 2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 32 x^{2} + 32 x^{2} + 4 - 4 \cdot u}}{2 x^{2}}$

Step 3.4.4.8.1.5.3.2

Add $32 x^{2}$ and $32 x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 64 x^{2} + 4 - 4 \cdot u}}{2 x^{2}}$

$y = \frac{16 x^{2} + 2 \pm \sqrt{256 x^{4} + 64 x^{2} + 4 - 4 u}}{2 x^{2}}$

Step 3.4.4.8.1.6

Factor $4$ out of $256 x^{4} + 64 x^{2} + 4 - 4 u$ .

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Step 3.4.4.8.1.6.1

Factor $4$ out of $256 x^{4}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 64 x^{2} + 4 - 4 u}}{2 x^{2}}$

Step 3.4.4.8.1.6.2

Factor $4$ out of $64 x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 4 (16 x^{2}) + 4 - 4 u}}{2 x^{2}}$

Step 3.4.4.8.1.6.3

Factor $4$ out of $4$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 4 (16 x^{2}) + 4 (1) - 4 u}}{2 x^{2}}$

Step 3.4.4.8.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4}) + 4 (16 x^{2}) + 4 (1) + 4 (- u)}}{2 x^{2}}$

Step 3.4.4.8.1.6.5

Factor $4$ out of $4 (64 x^{4}) + 4 (16 x^{2})$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2}) + 4 (1) + 4 (- u)}}{2 x^{2}}$

Step 3.4.4.8.1.6.6

Factor $4$ out of $4 (64 x^{4} + 16 x^{2}) + 4 (1)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1) + 4 (- u)}}{2 x^{2}}$

Step 3.4.4.8.1.6.7

Factor $4$ out of $4 (64 x^{4} + 16 x^{2} + 1) + 4 (- u)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - u)}}{2 x^{2}}$

Step 3.4.4.8.1.7

Replace all occurrences of $u$ with $x^{2} \cdot (64 x^{2} - 3)$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (x^{2} \cdot (64 x^{2} - 3)))}}{2 x^{2}}$

Step 3.4.4.8.1.8

Simplify.

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Step 3.4.4.8.1.8.1

Simplify each term.

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Step 3.4.4.8.1.8.1.1

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (x^{2} (64 x^{2}) + x^{2} \cdot - 3))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{2} x^{2} + x^{2} \cdot - 3))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.3

Move $- 3$ to the left of $x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{2} x^{2} - 3 \cdot x^{2}))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.4.4.8.1.8.1.4.1

Move $x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 (x^{2} x^{2}) - 3 \cdot x^{2}))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{2 + 2} - 3 \cdot x^{2}))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.4.3

Add $2$ and $2$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{4} - 3 \cdot x^{2}))}}{2 x^{2}}$

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{4} - 3 x^{2}))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.5

Apply the distributive property.

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - (64 x^{4}) - (- 3 x^{2}))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.6

Multiply $64$ by $- 1$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - 64 x^{4} - (- 3 x^{2}))}}{2 x^{2}}$

Step 3.4.4.8.1.8.1.7

Multiply $- 3$ by $- 1$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (64 x^{4} + 16 x^{2} + 1 - 64 x^{4} + 3 x^{2})}}{2 x^{2}}$

Step 3.4.4.8.1.8.2

Combine the opposite terms in $64 x^{4} + 16 x^{2} + 1 - 64 x^{4} + 3 x^{2}$ .

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Step 3.4.4.8.1.8.2.1

Subtract $64 x^{4}$ from $64 x^{4}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (16 x^{2} + 1 + 0 + 3 x^{2})}}{2 x^{2}}$

Step 3.4.4.8.1.8.2.2

Add $16 x^{2} + 1$ and $0$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (16 x^{2} + 1 + 3 x^{2})}}{2 x^{2}}$

Step 3.4.4.8.1.8.3

Add $16 x^{2}$ and $3 x^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{4 (19 x^{2} + 1)}}{2 x^{2}}$

Step 3.4.4.8.1.9

Rewrite $4 (19 x^{2} + 1)$ as $2^{2} (19 x^{2} + 1^{2})$ .

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Step 3.4.4.8.1.9.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{16 x^{2} + 2 \pm \sqrt{2^{2} (19 x^{2} + 1)}}{2 x^{2}}$

Step 3.4.4.8.1.9.2

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

$y = \frac{16 x^{2} + 2 \pm \sqrt{2^{2} (19 x^{2} + 1^{2})}}{2 x^{2}}$

Step 3.4.4.8.1.10

Pull terms out from under the radical.

$y = \frac{16 x^{2} + 2 \pm 2 \sqrt{19 x^{2} + 1^{2}}}{2 x^{2}}$

Step 3.4.4.8.1.11

One to any power is one.

$y = \frac{16 x^{2} + 2 \pm 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.8.2

Change the $\pm$ to $-$ .

$y = \frac{16 x^{2} + 2 - 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.8.3

Cancel the common factor of $16 x^{2} + 2 - 2 \sqrt{19 x^{2} + 1}$ and $2$ .

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Step 3.4.4.8.3.1

Factor $2$ out of $16 x^{2}$ .

$y = \frac{2 (8 x^{2}) + 2 - 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.8.3.2

Factor $2$ out of $2$ .

$y = \frac{2 (8 x^{2}) + 2 \cdot 1 - 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.8.3.3

Factor $2$ out of $2 (8 x^{2}) + 2 (1)$ .

$y = \frac{2 (8 x^{2} + 1) - 2 \sqrt{19 x^{2} + 1}}{2 x^{2}}$

Step 3.4.4.8.3.4

Factor $2$ out of $- 2 \sqrt{19 x^{2} + 1}$ .

$y = \frac{2 (8 x^{2} + 1) + 2 (- \sqrt{19 x^{2} + 1})}{2 x^{2}}$

Step 3.4.4.8.3.5

Factor $2$ out of $2 (8 x^{2} + 1) + 2 (- \sqrt{19 x^{2} + 1})$ .

$y = \frac{2 (8 x^{2} + 1 - \sqrt{19 x^{2} + 1})}{2 x^{2}}$

Step 3.4.4.8.3.6

Cancel the common factors.

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Step 3.4.4.8.3.6.1

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

$y = \frac{2 (8 x^{2} + 1 - \sqrt{19 x^{2} + 1})}{2 (x^{2})}$

Step 3.4.4.8.3.6.2

Cancel the common factor.

$y = \frac{2 (8 x^{2} + 1 - \sqrt{19 x^{2} + 1})}{2 x^{2}}$

Step 3.4.4.8.3.6.3

Rewrite the expression.

$y = \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$

Step 3.4.4.9

The final answer is the combination of both solutions.

$y = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$

$y = \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$

Step 4

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

$f^{- 1} (x) = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}, \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$

Step 5

Verify if $f^{- 1} (x) = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}, \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$ is the inverse of $f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$ and $f^{- 1} (x) = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}, \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$ .

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Step 5.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Step 5.3

Find the domain of $\frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$ .

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Step 5.3.1

Set the radicand in $\sqrt{19 x^{2} + 1}$ greater than or equal to $0$ to find where the expression is defined.

$19 x^{2} + 1 \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Subtract $1$ from both sides of the inequality.

$19 x^{2} \geq - 1$

Step 5.3.2.2

Divide each term in $19 x^{2} \geq - 1$ by $19$ and simplify.

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Step 5.3.2.2.1

Divide each term in $19 x^{2} \geq - 1$ by $19$ .

$\frac{19 x^{2}}{19} \geq \frac{- 1}{19}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Cancel the common factor of $19$ .

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Step 5.3.2.2.2.1.1

Cancel the common factor.

$\frac{19 x^{2}}{19} \geq \frac{- 1}{19}$

Step 5.3.2.2.2.1.2

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

$x^{2} \geq \frac{- 1}{19}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Move the negative in front of the fraction.

$x^{2} \geq - \frac{1}{19}$

Step 5.3.2.3

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 5.3.3

Set the denominator in $\frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 5.3.4

Solve for $x$ .

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Step 5.3.4.1

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

$x = \pm \sqrt{0}$

Step 5.3.4.2

Simplify $\pm \sqrt{0}$ .

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Step 5.3.4.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.3.4.2.2

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

$x = \pm 0$

Step 5.3.4.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.3.5

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}, \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$ is not equal to the range of $f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$ , then $f^{- 1} (x) = \frac{8 x^{2} + 1 + \sqrt{19 x^{2} + 1}}{x^{2}}, \frac{8 x^{2} + 1 - \sqrt{19 x^{2} + 1}}{x^{2}}$ is not an inverse of $f (x) = \frac{\sqrt{2 x + 3}}{x - 8}$ .

There is no inverse

Step 6