Trigonometry Examples

$f (x) x^{2} + 2 x - 1$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y x \cdot x^{2} + 2 x - 1 = f$ .

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

Step 2.2

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

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.3

Add $2$ and $1$ .

$y x^{3} + 2 x - 1 = f$

Step 2.3

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

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Step 2.3.1

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

$y x^{3} - 1 = f - 2 x$

Step 2.3.2

Add $1$ to both sides of the equation.

$y x^{3} = f - 2 x + 1$

Step 2.4

Divide each term in $y x^{3} = f - 2 x + 1$ by $x^{3}$ and simplify.

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Step 2.4.1

Divide each term in $y x^{3} = f - 2 x + 1$ by $x^{3}$ .

$\frac{y x^{3}}{x^{3}} = \frac{f}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{1}{x^{3}}$

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.

$\frac{y x^{3}}{x^{3}} = \frac{f}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{1}{x^{3}}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{f}{x^{3}} + \frac{- 2 x}{x^{3}} + \frac{1}{x^{3}}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

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

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Step 2.4.3.1.1.1

Factor $x$ out of $- 2 x$ .

$y = \frac{f}{x^{3}} + \frac{x \cdot - 2}{x^{3}} + \frac{1}{x^{3}}$

Step 2.4.3.1.1.2

Cancel the common factors.

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Step 2.4.3.1.1.2.1

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

$y = \frac{f}{x^{3}} + \frac{x \cdot - 2}{x \cdot x^{2}} + \frac{1}{x^{3}}$

Step 2.4.3.1.1.2.2

Cancel the common factor.

$y = \frac{f}{x^{3}} + \frac{x \cdot - 2}{x \cdot x^{2}} + \frac{1}{x^{3}}$

Step 2.4.3.1.1.2.3

Rewrite the expression.

$y = \frac{f}{x^{3}} + \frac{- 2}{x^{2}} + \frac{1}{x^{3}}$

Step 2.4.3.1.2

Move the negative in front of the fraction.

$y = \frac{f}{x^{3}} - \frac{2}{x^{2}} + \frac{1}{x^{3}}$

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify terms.

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Step 4.2.3.1

Combine the numerators over the common denominator.

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

Step 4.2.3.2

Combine the opposite terms in $f (x) \cdot x^{2} + 2 x - 1 + 1$ .

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Step 4.2.3.2.1

Add $- 1$ and $1$ .

$f^{- 1} (f (x) x^{2} + 2 x - 1) = \frac{f (x) \cdot x^{2} + 2 x + 0}{x^{3}} + \frac{- 2}{x^{2}}$

Step 4.2.3.2.2

Add $f (x) \cdot x^{2} + 2 x$ and $0$ .

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

Step 4.2.3.3

Reorder factors in $f (x) x^{2} + 2 x$ .

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

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Step 4.2.4.1.1

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

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

Step 4.2.4.1.2

Factor $x$ out of $2 x$ .

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

Step 4.2.4.1.3

Factor $x$ out of $x (x f (x)) + x \cdot 2$ .

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

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

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

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

Step 4.2.4.2.2

Cancel the common factor.

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

Step 4.2.4.2.3

Rewrite the expression.

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

Step 4.2.4.3

Move the negative in front of the fraction.

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

Step 4.2.5

Simplify terms.

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Step 4.2.5.1

Combine the numerators over the common denominator.

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

Step 4.2.5.2

Combine the opposite terms in $x f (x) + 2 - 2$ .

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Step 4.2.5.2.1

Subtract $2$ from $2$ .

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

Step 4.2.5.2.2

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

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

Step 4.2.5.3

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

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Step 4.2.5.3.1

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

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

Step 4.2.5.3.2

Cancel the common factors.

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Step 4.2.5.3.2.1

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

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

Step 4.2.5.3.2.2

Cancel the common factor.

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

Step 4.2.5.3.2.3

Rewrite the expression.

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

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

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

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 $x^{2}$ .

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

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 $1$ .

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

Step 4.3.3.1.2.2

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

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

Step 4.3.3.1.3

Add $2$ and $1$ .

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

Step 4.3.3.2

Apply the distributive property.

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

Step 4.3.3.3

Simplify.

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Step 4.3.3.3.1

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

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Step 4.3.3.3.1.1

Cancel the common factor.

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

Step 4.3.3.3.1.2

Rewrite the expression.

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

Step 4.3.3.3.2

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

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Step 4.3.3.3.2.1

Move the leading negative in $- \frac{2}{x^{2}}$ into the numerator.

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

Step 4.3.3.3.2.2

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

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

Step 4.3.3.3.2.3

Cancel the common factor.

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

Step 4.3.3.3.2.4

Rewrite the expression.

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

Step 4.3.3.3.3

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

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Step 4.3.3.3.3.1

Cancel the common factor.

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

Step 4.3.3.3.3.2

Rewrite the expression.

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

Step 4.3.4

Combine the opposite terms in $f - 2 x + 1 + 2 x - 1$ .

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Step 4.3.4.1

Add $- 2 x$ and $2 x$ .

$f (\frac{f}{x^{3}} - \frac{2}{x^{2}} + \frac{1}{x^{3}}) = f + 0 + 1 - 1$

Step 4.3.4.2

Add $f$ and $0$ .

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

Step 4.3.4.3

Subtract $1$ from $1$ .

$f (\frac{f}{x^{3}} - \frac{2}{x^{2}} + \frac{1}{x^{3}}) = f + 0$

Step 4.3.4.4

Add $f$ and $0$ .

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

Step 4.4

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

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