Algebra Examples

$f^{- 1} (x) = \sqrt{x - 5}$

Step 1

Solve for $\sqrt{x - 5}$ .

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Step 1.1

Rewrite the equation as $\sqrt{x - 5} = f^{- 1} x$ .

$\sqrt{x - 5} = f^{- 1} x$

Step 1.2

Simplify $f^{- 1} x$ .

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Step 1.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt{x - 5} = \frac{1}{f} x$

Step 1.2.2

Combine $\frac{1}{f}$ and $x$ .

$\sqrt{x - 5} = \frac{x}{f}$

Step 2

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

${\sqrt{x - 5}}^{2} = {(\frac{x}{f})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 5}$ as ${(x - 5)}^{\frac{1}{2}}$ .

${({(x - 5)}^{\frac{1}{2}})}^{2} = {(\frac{x}{f})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(\frac{x}{f})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{2} \cdot 2} = {(\frac{x}{f})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x - 5)}^{1} = {(\frac{x}{f})}^{2}$

Step 3.2.1.2

Simplify.

$x - 5 = {(\frac{x}{f})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the product rule to $\frac{x}{f}$ .

$x - 5 = \frac{x^{2}}{f^{2}}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $f^{2}$ .

$(x - 5) f^{2} = \frac{x^{2}}{f^{2}} f^{2}$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Apply the distributive property.

$x f^{2} - 5 f^{2} = \frac{x^{2}}{f^{2}} f^{2}$

Step 4.2.1.1.2

Reorder $x$ and $f^{2}$ .

$f^{2} x - 5 f^{2} = \frac{x^{2}}{f^{2}} f^{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Cancel the common factor.

$f^{2} x - 5 f^{2} = \frac{x^{2}}{f^{2}} f^{2}$

Step 4.2.2.1.2

Rewrite the expression.

$f^{2} x - 5 f^{2} = x^{2}$

Step 4.3

Solve for $x$ .

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Step 4.3.1

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

$f^{2} x - 5 f^{2} - x^{2} = 0$

Step 4.3.2

Use the quadratic formula to find the solutions.

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

Step 4.3.3

Substitute the values $a = - 1$ , $b = f^{2}$ , and $c = - 5 f^{2}$ into the quadratic formula and solve for $x$ .

$\frac{- f^{2} \pm \sqrt{{(f^{2})}^{2} - 4 \cdot (- 1 \cdot (- 5 f^{2}))}}{2 \cdot - 1}$

Step 4.3.4

Simplify.

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Step 4.3.4.1

Simplify the numerator.

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Step 4.3.4.1.1

Multiply the exponents in ${(f^{2})}^{2}$ .

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Step 4.3.4.1.1.1

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

$x = \frac{- f^{2} \pm \sqrt{f^{2 \cdot 2} - 4 \cdot - 1 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.4.1.1.2

Multiply $2$ by $2$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} - 4 \cdot - 1 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 5$ .

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Step 4.3.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} + 4 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.4.1.2.2

Multiply $4$ by $- 5$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} - 20 f^{2}}}{2 \cdot - 1}$

Step 4.3.4.1.3

Factor $f^{2}$ out of $f^{4} - 20 f^{2}$ .

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Step 4.3.4.1.3.1

Factor $f^{2}$ out of $f^{4}$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} f^{2} - 20 f^{2}}}{2 \cdot - 1}$

Step 4.3.4.1.3.2

Factor $f^{2}$ out of $- 20 f^{2}$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} f^{2} + f^{2} \cdot - 20}}{2 \cdot - 1}$

Step 4.3.4.1.3.3

Factor $f^{2}$ out of $f^{2} f^{2} + f^{2} \cdot - 20$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} (f^{2} - 20)}}{2 \cdot - 1}$

Step 4.3.4.1.4

Pull terms out from under the radical.

$x = \frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{2 \cdot - 1}$

Step 4.3.4.2

Multiply $2$ by $- 1$ .

$x = \frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{- 2}$

Step 4.3.4.3

Simplify $\frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{- 2}$ .

$x = \frac{f^{2} \pm f \sqrt{f^{2} - 20}}{2}$

Step 4.3.5

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

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Step 4.3.5.1

Simplify the numerator.

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Step 4.3.5.1.1

Multiply the exponents in ${(f^{2})}^{2}$ .

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Step 4.3.5.1.1.1

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

$x = \frac{- f^{2} \pm \sqrt{f^{2 \cdot 2} - 4 \cdot - 1 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.5.1.1.2

Multiply $2$ by $2$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} - 4 \cdot - 1 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 5$ .

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Step 4.3.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} + 4 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.5.1.2.2

Multiply $4$ by $- 5$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} - 20 f^{2}}}{2 \cdot - 1}$

Step 4.3.5.1.3

Factor $f^{2}$ out of $f^{4} - 20 f^{2}$ .

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Step 4.3.5.1.3.1

Factor $f^{2}$ out of $f^{4}$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} f^{2} - 20 f^{2}}}{2 \cdot - 1}$

Step 4.3.5.1.3.2

Factor $f^{2}$ out of $- 20 f^{2}$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} f^{2} + f^{2} \cdot - 20}}{2 \cdot - 1}$

Step 4.3.5.1.3.3

Factor $f^{2}$ out of $f^{2} f^{2} + f^{2} \cdot - 20$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} (f^{2} - 20)}}{2 \cdot - 1}$

Step 4.3.5.1.4

Pull terms out from under the radical.

$x = \frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{2 \cdot - 1}$

Step 4.3.5.2

Multiply $2$ by $- 1$ .

$x = \frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{- 2}$

Step 4.3.5.3

Simplify $\frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{- 2}$ .

$x = \frac{f^{2} \pm f \sqrt{f^{2} - 20}}{2}$

Step 4.3.5.4

Change the $\pm$ to $+$ .

$x = \frac{f^{2} + f \sqrt{f^{2} - 20}}{2}$

Step 4.3.5.5

Factor $f$ out of $f^{2} + f \sqrt{f^{2} - 20}$ .

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Step 4.3.5.5.1

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

$x = \frac{f \cdot f + f \sqrt{f^{2} - 20}}{2}$

Step 4.3.5.5.2

Factor $f$ out of $f \sqrt{f^{2} - 20}$ .

$x = \frac{f \cdot f + f (\sqrt{f^{2} - 20})}{2}$

Step 4.3.5.5.3

Factor $f$ out of $f \cdot f + f (\sqrt{f^{2} - 20})$ .

$x = \frac{f (f + \sqrt{f^{2} - 20})}{2}$

Step 4.3.6

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

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Step 4.3.6.1

Simplify the numerator.

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Step 4.3.6.1.1

Multiply the exponents in ${(f^{2})}^{2}$ .

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Step 4.3.6.1.1.1

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

$x = \frac{- f^{2} \pm \sqrt{f^{2 \cdot 2} - 4 \cdot - 1 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.6.1.1.2

Multiply $2$ by $2$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} - 4 \cdot - 1 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.6.1.2

Multiply $- 4 \cdot - 1 \cdot - 5$ .

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Step 4.3.6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} + 4 \cdot (- 5 f^{2})}}{2 \cdot - 1}$

Step 4.3.6.1.2.2

Multiply $4$ by $- 5$ .

$x = \frac{- f^{2} \pm \sqrt{f^{4} - 20 f^{2}}}{2 \cdot - 1}$

Step 4.3.6.1.3

Factor $f^{2}$ out of $f^{4} - 20 f^{2}$ .

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Step 4.3.6.1.3.1

Factor $f^{2}$ out of $f^{4}$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} f^{2} - 20 f^{2}}}{2 \cdot - 1}$

Step 4.3.6.1.3.2

Factor $f^{2}$ out of $- 20 f^{2}$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} f^{2} + f^{2} \cdot - 20}}{2 \cdot - 1}$

Step 4.3.6.1.3.3

Factor $f^{2}$ out of $f^{2} f^{2} + f^{2} \cdot - 20$ .

$x = \frac{- f^{2} \pm \sqrt{f^{2} (f^{2} - 20)}}{2 \cdot - 1}$

Step 4.3.6.1.4

Pull terms out from under the radical.

$x = \frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{2 \cdot - 1}$

Step 4.3.6.2

Multiply $2$ by $- 1$ .

$x = \frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{- 2}$

Step 4.3.6.3

Simplify $\frac{- f^{2} \pm f \sqrt{f^{2} - 20}}{- 2}$ .

$x = \frac{f^{2} \pm f \sqrt{f^{2} - 20}}{2}$

Step 4.3.6.4

Change the $\pm$ to $-$ .

$x = \frac{f^{2} - f \sqrt{f^{2} - 20}}{2}$

Step 4.3.6.5

Factor $f$ out of $f^{2} - f \sqrt{f^{2} - 20}$ .

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Step 4.3.6.5.1

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

$x = \frac{f \cdot f - f \sqrt{f^{2} - 20}}{2}$

Step 4.3.6.5.2

Factor $f$ out of $- f \sqrt{f^{2} - 20}$ .

$x = \frac{f \cdot f + f (- \sqrt{f^{2} - 20})}{2}$

Step 4.3.6.5.3

Factor $f$ out of $f \cdot f + f (- \sqrt{f^{2} - 20})$ .

$x = \frac{f (f - \sqrt{f^{2} - 20})}{2}$

Step 4.3.7

The final answer is the combination of both solutions.

$x = \frac{f (f + \sqrt{f^{2} - 20})}{2}$

$x = \frac{f (f - \sqrt{f^{2} - 20})}{2}$

$x = \frac{f (f + \sqrt{f^{2} - 20})}{2}$

$x = \frac{f (f - \sqrt{f^{2} - 20})}{2}$

$x = \frac{f (f + \sqrt{f^{2} - 20})}{2}$

$x = \frac{f (f - \sqrt{f^{2} - 20})}{2}$

Step 5

To rewrite as a function of $f$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $f$ is on the other side.

$f (f) = \frac{f (f + \sqrt{f^{2} - 20})}{2}, \frac{f (f - \sqrt{f^{2} - 20})}{2}$