Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y x - \sqrt{3 x - 2} - 1 = f$ .

$y x - \sqrt{3 x - 2} - 1 = f$

Step 2.2

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

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Step 2.2.1

Add $\sqrt{3 x - 2}$ to both sides of the equation.

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

Step 2.2.2

Add $1$ to both sides of the equation.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $y$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Combine the opposite terms in $\frac{(f (x) - \sqrt{3 x - 2} - 1) + \sqrt{3 x - 2}}{x} + \frac{1}{x}$ .

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Step 4.2.3.1

Add $- \sqrt{3 x - 2}$ and $\sqrt{3 x - 2}$ .

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

Step 4.2.3.2

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Simplify by adding numbers.

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Step 4.2.5.1

Add $- 1$ and $1$ .

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

Step 4.2.5.2

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

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

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Cancel the common factor of $x$ .

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Step 4.3.3.2.1

Cancel the common factor.

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

Step 4.3.3.2.2

Rewrite the expression.

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

Step 4.3.3.3

Cancel the common factor of $x$ .

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Step 4.3.3.3.1

Cancel the common factor.

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

Step 4.3.3.3.2

Rewrite the expression.

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

Step 4.3.4

Combine the opposite terms in $f + \sqrt{3 x - 2} + 1 - \sqrt{3 x - 2} - 1$ .

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Step 4.3.4.1

Subtract $\sqrt{3 x - 2}$ from $\sqrt{3 x - 2}$ .

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

Step 4.3.4.2

Add $f$ and $0$ .

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

Step 4.3.4.3

Subtract $1$ from $1$ .

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

Step 4.3.4.4

Add $f$ and $0$ .

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

Step 4.4

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

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