Algebra Examples

$f (x) = - 9 \sqrt{x - 8} + 5$ , $x \geq 8$

Step 1

Find the range of the given function.

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Step 1.1

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

$(- \infty, 5]$

Step 1.2

Convert $(- \infty, 5]$ to an inequality.

$y \leq 5$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

$x = - 9 \sqrt{y - 8} + 5$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $- 9 \sqrt{y - 8} + 5 = x$ .

$- 9 \sqrt{y - 8} + 5 = x$

Step 2.2.2

Subtract $5$ from both sides of the equation.

$- 9 \sqrt{y - 8} = x - 5$

Step 2.2.3

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

${(- 9 \sqrt{y - 8})}^{2} = {(x - 5)}^{2}$

Step 2.2.4

Simplify each side of the equation.

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Step 2.2.4.1

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

${(- 9 {(y - 8)}^{\frac{1}{2}})}^{2} = {(x - 5)}^{2}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Simplify ${(- 9 {(y - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.4.2.1.1

Apply the product rule to $- 9 {(y - 8)}^{\frac{1}{2}}$ .

${(- 9)}^{2} {({(y - 8)}^{\frac{1}{2}})}^{2} = {(x - 5)}^{2}$

Step 2.2.4.2.1.2

Raise $- 9$ to the power of $2$ .

$81 {({(y - 8)}^{\frac{1}{2}})}^{2} = {(x - 5)}^{2}$

Step 2.2.4.2.1.3

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

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Step 2.2.4.2.1.3.1

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

$81 {(y - 8)}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Step 2.2.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.3.2.1

Cancel the common factor.

$81 {(y - 8)}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Step 2.2.4.2.1.3.2.2

Rewrite the expression.

$81 {(y - 8)}^{1} = {(x - 5)}^{2}$

Step 2.2.4.2.1.4

Simplify.

$81 (y - 8) = {(x - 5)}^{2}$

Step 2.2.4.2.1.5

Apply the distributive property.

$81 y + 81 \cdot - 8 = {(x - 5)}^{2}$

Step 2.2.4.2.1.6

Multiply $81$ by $- 8$ .

$81 y - 648 = {(x - 5)}^{2}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Simplify ${(x - 5)}^{2}$ .

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Step 2.2.4.3.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$81 y - 648 = (x - 5) (x - 5)$

Step 2.2.4.3.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.2.4.3.1.2.1

Apply the distributive property.

$81 y - 648 = x (x - 5) - 5 (x - 5)$

Step 2.2.4.3.1.2.2

Apply the distributive property.

$81 y - 648 = x \cdot x + x \cdot - 5 - 5 (x - 5)$

Step 2.2.4.3.1.2.3

Apply the distributive property.

$81 y - 648 = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$

Step 2.2.4.3.1.3

Simplify and combine like terms.

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Step 2.2.4.3.1.3.1

Simplify each term.

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Step 2.2.4.3.1.3.1.1

Multiply $x$ by $x$ .

$81 y - 648 = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$

Step 2.2.4.3.1.3.1.2

Move $- 5$ to the left of $x$ .

$81 y - 648 = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$

Step 2.2.4.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$81 y - 648 = x^{2} - 5 x - 5 x + 25$

Step 2.2.4.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$81 y - 648 = x^{2} - 10 x + 25$

Step 2.2.5

Solve for $y$ .

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Step 2.2.5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.5.1.1

Add $648$ to both sides of the equation.

$81 y = x^{2} - 10 x + 25 + 648$

Step 2.2.5.1.2

Add $25$ and $648$ .

$81 y = x^{2} - 10 x + 673$

Step 2.2.5.2

Divide each term in $81 y = x^{2} - 10 x + 673$ by $81$ and simplify.

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Step 2.2.5.2.1

Divide each term in $81 y = x^{2} - 10 x + 673$ by $81$ .

$\frac{81 y}{81} = \frac{x^{2}}{81} + \frac{- 10 x}{81} + \frac{673}{81}$

Step 2.2.5.2.2

Simplify the left side.

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Step 2.2.5.2.2.1

Cancel the common factor of $81$ .

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Step 2.2.5.2.2.1.1

Cancel the common factor.

$\frac{81 y}{81} = \frac{x^{2}}{81} + \frac{- 10 x}{81} + \frac{673}{81}$

Step 2.2.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{81} + \frac{- 10 x}{81} + \frac{673}{81}$

Step 2.2.5.2.3

Simplify the right side.

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Step 2.2.5.2.3.1

Move the negative in front of the fraction.

$y = \frac{x^{2}}{81} - \frac{10 x}{81} + \frac{673}{81}$

Step 2.3

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

$f^{- 1} (x) = \frac{x^{2}}{81} - \frac{10 x}{81} + \frac{673}{81}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = \frac{x^{2}}{81} - \frac{10 x}{81} + \frac{673}{81}, x \leq 5$

Step 4