Algebra Examples

$f^{- 1} (x) = \sqrt{x - 3}$

Step 1

Divide each term in $f^{- 1} x = \sqrt{x - 3}$ by $x$ and simplify.

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Step 1.1

Divide each term in $f^{- 1} x = \sqrt{x - 3}$ by $x$ .

$\frac{f^{- 1} x}{x} = \frac{\sqrt{x - 3}}{x}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Move $f^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x}{x f} = \frac{\sqrt{x - 3}}{x}$

Step 1.2.2

Cancel the common factor of $x$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{x}{x f} = \frac{\sqrt{x - 3}}{x}$

Step 1.2.2.2

Rewrite the expression.

$\frac{1}{f} = \frac{\sqrt{x - 3}}{x}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 x = f \sqrt{x - 3}$

Step 3

Solve the equation for $f$ .

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Step 3.1

Rewrite the equation as $f \sqrt{x - 3} = 1 x$ .

$f \sqrt{x - 3} = 1 x$

Step 3.2

Multiply $x$ by $1$ .

$f \sqrt{x - 3} = x$

Step 3.3

Divide each term in $f \sqrt{x - 3} = x$ by $\sqrt{x - 3}$ and simplify.

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Step 3.3.1

Divide each term in $f \sqrt{x - 3} = x$ by $\sqrt{x - 3}$ .

$\frac{f \sqrt{x - 3}}{\sqrt{x - 3}} = \frac{x}{\sqrt{x - 3}}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $\sqrt{x - 3}$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{f \sqrt{x - 3}}{\sqrt{x - 3}} = \frac{x}{\sqrt{x - 3}}$

Step 3.3.2.1.2

Divide $f$ by $1$ .

$f = \frac{x}{\sqrt{x - 3}}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $\frac{x}{\sqrt{x - 3}}$ by $\frac{\sqrt{x - 3}}{\sqrt{x - 3}}$ .

$f = \frac{x}{\sqrt{x - 3}} \cdot \frac{\sqrt{x - 3}}{\sqrt{x - 3}}$

Step 3.3.3.2

Combine and simplify the denominator.

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Step 3.3.3.2.1

Multiply $\frac{x}{\sqrt{x - 3}}$ by $\frac{\sqrt{x - 3}}{\sqrt{x - 3}}$ .

$f = \frac{x \sqrt{x - 3}}{\sqrt{x - 3} \sqrt{x - 3}}$

Step 3.3.3.2.2

Raise $\sqrt{x - 3}$ to the power of $1$ .

$f = \frac{x \sqrt{x - 3}}{{\sqrt{x - 3}}^{1} \sqrt{x - 3}}$

Step 3.3.3.2.3

Raise $\sqrt{x - 3}$ to the power of $1$ .

$f = \frac{x \sqrt{x - 3}}{{\sqrt{x - 3}}^{1} {\sqrt{x - 3}}^{1}}$

Step 3.3.3.2.4

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

$f = \frac{x \sqrt{x - 3}}{{\sqrt{x - 3}}^{1 + 1}}$

Step 3.3.3.2.5

Add $1$ and $1$ .

$f = \frac{x \sqrt{x - 3}}{{\sqrt{x - 3}}^{2}}$

Step 3.3.3.2.6

Rewrite ${\sqrt{x - 3}}^{2}$ as $x - 3$ .

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Step 3.3.3.2.6.1

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

$f = \frac{x \sqrt{x - 3}}{{({(x - 3)}^{\frac{1}{2}})}^{2}}$

Step 3.3.3.2.6.2

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

$f = \frac{x \sqrt{x - 3}}{{(x - 3)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.3.2.6.3

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

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

Step 3.3.3.2.6.4

Cancel the common factor of $2$ .

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Step 3.3.3.2.6.4.1

Cancel the common factor.

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

Step 3.3.3.2.6.4.2

Rewrite the expression.

$f = \frac{x \sqrt{x - 3}}{{(x - 3)}^{1}}$

Step 3.3.3.2.6.5

Simplify.

$f = \frac{x \sqrt{x - 3}}{x - 3}$

Step 4

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

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