Precalculus Examples

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

Step 1

Write $f (x) = - 1 - \frac{3}{4} x$ as an equation.

$y = - 1 - \frac{3}{4} x$

Step 2

Interchange the variables.

$x = - 1 - \frac{3}{4} y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 1 - \frac{3}{4} y = x$ .

$- 1 - \frac{3}{4} y = x$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $y$ and $\frac{3}{4}$ .

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

Step 3.2.2

Move $3$ to the left of $y$ .

$- 1 - \frac{3 y}{4} = x$

Step 3.3

Add $1$ to both sides of the equation.

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

Step 3.4

Multiply both sides of the equation by $- \frac{4}{3}$ .

$- \frac{4}{3} (- \frac{3 y}{4}) = - \frac{4}{3} (x + 1)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $- \frac{4}{3} (- \frac{3 y}{4})$ .

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Step 3.5.1.1.1

Cancel the common factor of $4$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} (- \frac{3 y}{4}) = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{3 y}{4}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.2

Cancel the common factor of $3$ .

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Step 3.5.1.1.2.1

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.3

Multiply.

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Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{4}{3} (x + 1)$

Step 3.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{4}{3} (x + 1)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- \frac{4}{3} (x + 1)$ .

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Step 3.5.2.1.1

Simplify terms.

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Step 3.5.2.1.1.1

Apply the distributive property.

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

Step 3.5.2.1.1.2

Combine $x$ and $\frac{4}{3}$ .

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

Step 3.5.2.1.1.3

Multiply $- 1$ by $1$ .

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

Step 3.5.2.1.2

Move $4$ to the left of $x$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Simplify each term.

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Step 5.2.4.1.1

Combine $x$ and $\frac{3}{4}$ .

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

Step 5.2.4.1.2

Move $3$ to the left of $x$ .

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

Step 5.2.4.2

Apply the distributive property.

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

Step 5.2.4.3

Multiply $- 4$ by $- 1$ .

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

Step 5.2.4.4

Cancel the common factor of $4$ .

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Step 5.2.4.4.1

Move the leading negative in $- \frac{3 x}{4}$ into the numerator.

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

Step 5.2.4.4.2

Factor $4$ out of $- 4$ .

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

Step 5.2.4.4.3

Cancel the common factor.

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

Step 5.2.4.4.4

Rewrite the expression.

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

Step 5.2.4.5

Multiply $- 3$ by $- 1$ .

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

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $4 + 3 x - 4$ .

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Step 5.2.5.1.1

Subtract $4$ from $4$ .

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

Step 5.2.5.1.2

Add $3 x$ and $0$ .

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

Step 5.2.5.2

Cancel the common factor of $3$ .

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Step 5.2.5.2.1

Cancel the common factor.

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

Step 5.2.5.2.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

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

Step 5.3.3.2.2

Move the leading negative in $- \frac{4 x}{3}$ into the numerator.

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

Step 5.3.3.2.3

Factor $3$ out of $- 3$ .

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

Step 5.3.3.2.4

Cancel the common factor.

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

Step 5.3.3.2.5

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $4$ .

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Step 5.3.3.3.1

Factor $4$ out of $- 4 x$ .

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

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

Step 5.3.3.4

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.5

Multiply $x$ by $1$ .

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

Step 5.3.3.6

Cancel the common factor of $3$ .

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Step 5.3.3.6.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

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

Step 5.3.3.6.2

Move the leading negative in $- \frac{4}{3}$ into the numerator.

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

Step 5.3.3.6.3

Factor $3$ out of $- 3$ .

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

Step 5.3.3.6.4

Cancel the common factor.

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

Step 5.3.3.6.5

Rewrite the expression.

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

Step 5.3.3.7

Cancel the common factor of $4$ .

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Step 5.3.3.7.1

Factor $4$ out of $- 4$ .

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

Step 5.3.3.7.2

Cancel the common factor.

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

Step 5.3.3.7.3

Rewrite the expression.

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

Step 5.3.3.8

Multiply $- 1$ by $- 1$ .

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

Step 5.3.4

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

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Step 5.3.4.1

Add $- 1$ and $1$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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