Precalculus Examples

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

Step 1

Write $f (x) = 2 x + \frac{1}{3}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 y + \frac{1}{3} = x$ .

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

Step 3.2

Subtract $\frac{1}{3}$ from both sides of the equation.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.3.1.2

Multiply $- \frac{1}{3} \cdot \frac{1}{2}$ .

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Step 3.3.3.1.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 3.3.3.1.2.2

Multiply $2$ by $3$ .

$y = \frac{x}{2} - \frac{1}{6}$

Step 4

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

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

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2.3.1.2

Combine $2 x$ and $\frac{3}{3}$ .

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

Step 5.2.3.1.3

Combine the numerators over the common denominator.

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

Step 5.2.3.1.4

Multiply $3$ by $2$ .

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

Step 5.2.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.3.3

Multiply $\frac{6 x + 1}{3} \cdot \frac{1}{2}$ .

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Step 5.2.3.3.1

Multiply $\frac{6 x + 1}{3}$ by $\frac{1}{2}$ .

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

Step 5.2.3.3.2

Multiply $3$ by $2$ .

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

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

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

Step 5.2.4.2

Combine the opposite terms in $6 x + 1 - 1$ .

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Step 5.2.4.2.1

Subtract $1$ from $1$ .

$f^{- 1} (2 x + \frac{1}{3}) = \frac{6 x + 0}{6}$

Step 5.2.4.2.2

Add $6 x$ and $0$ .

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

Step 5.2.4.3

Cancel the common factor of $6$ .

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Step 5.2.4.3.1

Cancel the common factor.

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

Step 5.2.4.3.2

Divide $x$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

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

Step 5.3.3.3.2

Factor $2$ out of $6$ .

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

Step 5.3.3.3.3

Cancel the common factor.

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

Step 5.3.3.3.4

Rewrite the expression.

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

Step 5.3.3.4

Move the negative in front of the fraction.

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

Step 5.3.4

Combine the opposite terms in $x - \frac{1}{3} + \frac{1}{3}$ .

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Step 5.3.4.1

Add $- \frac{1}{3}$ and $\frac{1}{3}$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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