Precalculus Examples

$f (x) = 2 x - \frac{3}{4}$

Step 1

Write $f (x) = 2 x - \frac{3}{4}$ as an equation.

$y = 2 x - \frac{3}{4}$

Step 2

Interchange the variables.

$x = 2 y - \frac{3}{4}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 y - \frac{3}{4} = x$ .

$2 y - \frac{3}{4} = x$

Step 3.2

Add $\frac{3}{4}$ to both sides of the equation.

$2 y = x + \frac{3}{4}$

Step 3.3

Divide each term in $2 y = x + \frac{3}{4}$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 y = x + \frac{3}{4}$ by $2$ .

$\frac{2 y}{2} = \frac{x}{2} + \frac{\frac{3}{4}}{2}$

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.

$\frac{2 y}{2} = \frac{x}{2} + \frac{\frac{3}{4}}{2}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{2} + \frac{\frac{3}{4}}{2}$

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.

$y = \frac{x}{2} + \frac{3}{4} \cdot \frac{1}{2}$

Step 3.3.3.1.2

Multiply $\frac{3}{4} \cdot \frac{1}{2}$ .

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Step 3.3.3.1.2.1

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

$y = \frac{x}{2} + \frac{3}{4 \cdot 2}$

Step 3.3.3.1.2.2

Multiply $4$ by $2$ .

$y = \frac{x}{2} + \frac{3}{8}$

Step 4

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

$f^{- 1} (x) = \frac{x}{2} + \frac{3}{8}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{2} + \frac{3}{8}$ is the inverse of $f (x) = 2 x - \frac{3}{4}$ .

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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 $f^{- 1} (2 x - \frac{3}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{2 x - \frac{3}{4}}{2} + \frac{3}{8}$

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 $\frac{4}{4}$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{2 x \cdot \frac{4}{4} - \frac{3}{4}}{2} + \frac{3}{8}$

Step 5.2.3.1.2

Combine $2 x$ and $\frac{4}{4}$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{\frac{2 x \cdot 4}{4} - \frac{3}{4}}{2} + \frac{3}{8}$

Step 5.2.3.1.3

Combine the numerators over the common denominator.

$f^{- 1} (2 x - \frac{3}{4}) = \frac{\frac{2 x \cdot 4 - 3}{4}}{2} + \frac{3}{8}$

Step 5.2.3.1.4

Multiply $4$ by $2$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{\frac{8 x - 3}{4}}{2} + \frac{3}{8}$

Step 5.2.3.2

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x - 3}{4} \cdot \frac{1}{2} + \frac{3}{8}$

Step 5.2.3.3

Multiply $\frac{8 x - 3}{4} \cdot \frac{1}{2}$ .

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Step 5.2.3.3.1

Multiply $\frac{8 x - 3}{4}$ by $\frac{1}{2}$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x - 3}{4 \cdot 2} + \frac{3}{8}$

Step 5.2.3.3.2

Multiply $4$ by $2$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x - 3}{8} + \frac{3}{8}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x - 3 + 3}{8}$

Step 5.2.4.2

Combine the opposite terms in $8 x - 3 + 3$ .

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Step 5.2.4.2.1

Add $- 3$ and $3$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x + 0}{8}$

Step 5.2.4.2.2

Add $8 x$ and $0$ .

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x}{8}$

Step 5.2.4.3

Cancel the common factor of $8$ .

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Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (2 x - \frac{3}{4}) = \frac{8 x}{8}$

Step 5.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (2 x - \frac{3}{4}) = x$

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 $f (\frac{x}{2} + \frac{3}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{2} + \frac{3}{8}) = 2 (\frac{x}{2} + \frac{3}{8}) - \frac{3}{4}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{2} + \frac{3}{8}) = 2 (\frac{x}{2}) + 2 (\frac{3}{8}) - \frac{3}{4}$

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.

$f (\frac{x}{2} + \frac{3}{8}) = 2 (\frac{x}{2}) + 2 (\frac{3}{8}) - \frac{3}{4}$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{x}{2} + \frac{3}{8}) = x + 2 (\frac{3}{8}) - \frac{3}{4}$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Factor $2$ out of $8$ .

$f (\frac{x}{2} + \frac{3}{8}) = x + 2 (\frac{3}{2 (4)}) - \frac{3}{4}$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{x}{2} + \frac{3}{8}) = x + 2 (\frac{3}{2 \cdot 4}) - \frac{3}{4}$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{x}{2} + \frac{3}{8}) = x + \frac{3}{4} - \frac{3}{4}$

Step 5.3.4

Combine the opposite terms in $x + \frac{3}{4} - \frac{3}{4}$ .

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Step 5.3.4.1

Combine the numerators over the common denominator.

$f (\frac{x}{2} + \frac{3}{8}) = x + \frac{3 - 3}{4}$

Step 5.3.4.2

Subtract $3$ from $3$ .

$f (\frac{x}{2} + \frac{3}{8}) = x + \frac{0}{4}$

Step 5.3.4.3

Divide $0$ by $4$ .

$f (\frac{x}{2} + \frac{3}{8}) = x + 0$

Step 5.3.4.4

Add $x$ and $0$ .

$f (\frac{x}{2} + \frac{3}{8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{2} + \frac{3}{8}$ is the inverse of $f (x) = 2 x - \frac{3}{4}$ .

$f^{- 1} (x) = \frac{x}{2} + \frac{3}{8}$