Finite Math Examples

$f (x) x^{2} - 6 x$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y x \cdot x^{2} - 6 x = f$ .

$y x \cdot x^{2} - 6 x = f$

Step 2.2

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

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

Multiply $x^{2}$ by $x$ .

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Step 2.2.2.1

Raise $x$ to the power of $1$ .

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

Step 2.2.2.2

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

$y x^{2 + 1} - 6 x = f$

Step 2.2.3

Add $2$ and $1$ .

$y x^{3} - 6 x = f$

Step 2.3

Add $6 x$ to both sides of the equation.

$y x^{3} = f + 6 x$

Step 2.4

Divide each term in $y x^{3} = f + 6 x$ by $x^{3}$ and simplify.

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Step 2.4.1

Divide each term in $y x^{3} = f + 6 x$ by $x^{3}$ .

$\frac{y x^{3}}{x^{3}} = \frac{f}{x^{3}} + \frac{6 x}{x^{3}}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{y x^{3}}{x^{3}} = \frac{f}{x^{3}} + \frac{6 x}{x^{3}}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{f}{x^{3}} + \frac{6 x}{x^{3}}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $x$ and $x^{3}$ .

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Step 2.4.3.1.1

Factor $x$ out of $6 x$ .

$y = \frac{f}{x^{3}} + \frac{x \cdot 6}{x^{3}}$

Step 2.4.3.1.2

Cancel the common factors.

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Step 2.4.3.1.2.1

Factor $x$ out of $x^{3}$ .

$y = \frac{f}{x^{3}} + \frac{x \cdot 6}{x \cdot x^{2}}$

Step 2.4.3.1.2.2

Cancel the common factor.

$y = \frac{f}{x^{3}} + \frac{x \cdot 6}{x \cdot x^{2}}$

Step 2.4.3.1.2.3

Rewrite the expression.

$y = \frac{f}{x^{3}} + \frac{6}{x^{2}}$

Step 3

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

$f^{- 1} (f) = \frac{f}{x^{3}} + \frac{6}{x^{2}}$

Step 4

Verify if $f^{- 1} (f) = \frac{f}{x^{3}} + \frac{6}{x^{2}}$ is the inverse of $f (f) = f (x) x^{2} - 6 x$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (f (x) x^{2} - 6 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (f (x) x^{2} - 6 x) = \frac{f (x) \cdot x^{2} - 6 x}{x^{3}} + \frac{6}{x^{2}}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Factor $x$ out of $f (x) \cdot x^{2} - 6 x$ .

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Step 4.2.3.1.1

Factor $x$ out of $f (x) \cdot x^{2}$ .

$f^{- 1} (f (x) x^{2} - 6 x) = \frac{x (f (x) \cdot x) - 6 x}{x^{3}} + \frac{6}{x^{2}}$

Step 4.2.3.1.2

Factor $x$ out of $- 6 x$ .

$f^{- 1} (f (x) x^{2} - 6 x) = \frac{x (f (x) \cdot x) + x \cdot - 6}{x^{3}} + \frac{6}{x^{2}}$

Step 4.2.3.1.3

Factor $x$ out of $x (f (x) \cdot x) + x \cdot - 6$ .

$f^{- 1} (f (x) x^{2} - 6 x) = \frac{x (f (x) \cdot x - 6)}{x^{3}} + \frac{6}{x^{2}}$

$f^{- 1} (f (x) x^{2} - 6 x) = \frac{x (f (x) x - 6)}{x^{3}} + \frac{6}{x^{2}}$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $x$ out of $x^{3}$ .

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

Step 4.2.3.2.2

Cancel the common factor.

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

Step 4.2.3.2.3

Rewrite the expression.

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

Step 4.2.4

Simplify terms.

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Step 4.2.4.1

Combine the numerators over the common denominator.

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

Step 4.2.4.2

Combine the opposite terms in $f (x) x - 6 + 6$ .

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Step 4.2.4.2.1

Add $- 6$ and $6$ .

$f^{- 1} (f (x) x^{2} - 6 x) = \frac{f (x) x + 0}{x^{2}}$

Step 4.2.4.2.2

Add $f (x) x$ and $0$ .

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

Step 4.2.4.3

Reorder factors in $f (x) x$ .

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

Step 4.2.4.4

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

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Step 4.2.4.4.1

Factor $x$ out of $x f (x)$ .

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

Step 4.2.4.4.2

Cancel the common factors.

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Step 4.2.4.4.2.1

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

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

Step 4.2.4.4.2.2

Cancel the common factor.

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

Step 4.2.4.4.2.3

Rewrite the expression.

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{f}{x^{3}} + \frac{6}{x^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = (\frac{f}{x^{3}} + \frac{6}{x^{2}}) (x) \cdot x^{2} - 6 x$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

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

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Step 4.3.3.1.1

Move $x^{2}$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = (\frac{f}{x^{3}} + \frac{6}{x^{2}}) (x^{2} x) - 6 x$

Step 4.3.3.1.2

Multiply $x^{2}$ by $x$ .

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Step 4.3.3.1.2.1

Raise $x$ to the power of $1$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = (\frac{f}{x^{3}} + \frac{6}{x^{2}}) (x^{2} x) - 6 x$

Step 4.3.3.1.2.2

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

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = (\frac{f}{x^{3}} + \frac{6}{x^{2}}) x^{2 + 1} - 6 x$

Step 4.3.3.1.3

Add $2$ and $1$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = (\frac{f}{x^{3}} + \frac{6}{x^{2}}) x^{3} - 6 x$

Step 4.3.3.2

Apply the distributive property.

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = \frac{f}{x^{3}} \cdot x^{3} + \frac{6}{x^{2}} \cdot x^{3} - 6 x$

Step 4.3.3.3

Cancel the common factor of $x^{3}$ .

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Step 4.3.3.3.1

Cancel the common factor.

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = \frac{f}{x^{3}} \cdot x^{3} + \frac{6}{x^{2}} \cdot x^{3} - 6 x$

Step 4.3.3.3.2

Rewrite the expression.

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = f + \frac{6}{x^{2}} \cdot x^{3} - 6 x$

Step 4.3.3.4

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

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Step 4.3.3.4.1

Factor $x^{2}$ out of $x^{3}$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = f + \frac{6}{x^{2}} \cdot (x^{2} x) - 6 x$

Step 4.3.3.4.2

Cancel the common factor.

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = f + \frac{6}{x^{2}} \cdot (x^{2} x) - 6 x$

Step 4.3.3.4.3

Rewrite the expression.

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = f + 6 x - 6 x$

Step 4.3.4

Combine the opposite terms in $f + 6 x - 6 x$ .

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Step 4.3.4.1

Subtract $6 x$ from $6 x$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = f + 0$

Step 4.3.4.2

Add $f$ and $0$ .

$f (\frac{f}{x^{3}} + \frac{6}{x^{2}}) = f$

Step 4.4

Since $f^{- 1} (f (f)) = f$ and $f (f^{- 1} (f)) = f$ , then $f^{- 1} (f) = \frac{f}{x^{3}} + \frac{6}{x^{2}}$ is the inverse of $f (f) = f (x) x^{2} - 6 x$ .

$f^{- 1} (f) = \frac{f}{x^{3}} + \frac{6}{x^{2}}$