Precalculus Examples

$g (x) = \frac{- 4 x - 4}{5}$

Step 1

Write $g (x) = \frac{- 4 x - 4}{5}$ as an equation.

$y = \frac{- 4 x - 4}{5}$

Step 2

Interchange the variables.

$x = \frac{- 4 y - 4}{5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{- 4 y - 4}{5} = x$ .

$\frac{- 4 y - 4}{5} = x$

Step 3.2

Multiply both sides by $5$ .

$\frac{- 4 y - 4}{5} \cdot 5 = x \cdot 5$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{- 4 y - 4}{5} \cdot 5$ .

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Step 3.3.1.1.1

Factor $4$ out of $- 4 y - 4$ .

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Step 3.3.1.1.1.1

Factor $4$ out of $- 4 y$ .

$\frac{4 (- y) - 4}{5} \cdot 5 = x \cdot 5$

Step 3.3.1.1.1.2

Factor $4$ out of $- 4$ .

$\frac{4 (- y) + 4 (- 1)}{5} \cdot 5 = x \cdot 5$

Step 3.3.1.1.1.3

Factor $4$ out of $4 (- y) + 4 (- 1)$ .

$\frac{4 (- y - 1)}{5} \cdot 5 = x \cdot 5$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{4 (- y - 1)}{5} \cdot 5 = x \cdot 5$

Step 3.3.1.1.2.2

Rewrite the expression.

$4 (- y - 1) = x \cdot 5$

Step 3.3.1.1.3

Apply the distributive property.

$4 (- y) + 4 \cdot - 1 = x \cdot 5$

Step 3.3.1.1.4

Multiply.

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Step 3.3.1.1.4.1

Multiply $- 1$ by $4$ .

$- 4 y + 4 \cdot - 1 = x \cdot 5$

Step 3.3.1.1.4.2

Multiply $4$ by $- 1$ .

$- 4 y - 4 = x \cdot 5$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move $5$ to the left of $x$ .

$- 4 y - 4 = 5 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $4$ to both sides of the equation.

$- 4 y = 5 x + 4$

Step 3.4.2

Divide each term in $- 4 y = 5 x + 4$ by $- 4$ and simplify.

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Step 3.4.2.1

Divide each term in $- 4 y = 5 x + 4$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{5 x}{- 4} + \frac{4}{- 4}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{5 x}{- 4} + \frac{4}{- 4}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{- 4} + \frac{4}{- 4}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{5 x}{4} + \frac{4}{- 4}$

Step 3.4.2.3.1.2

Divide $4$ by $- 4$ .

$y = - \frac{5 x}{4} - 1$

Step 4

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

$g^{- 1} (x) = - \frac{5 x}{4} - 1$

Step 5

Verify if $g^{- 1} (x) = - \frac{5 x}{4} - 1$ is the inverse of $g (x) = \frac{- 4 x - 4}{5}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $g^{- 1} (\frac{- 4 x - 4}{5})$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{5 (\frac{- 4 x - 4}{5})}{4} - 1$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

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

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{5 (\frac{4 (- x) - 4}{5})}{4} - 1$

Step 5.2.3.1.2

Factor $4$ out of $- 4$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{5 (\frac{4 (- x) + 4 (- 1)}{5})}{4} - 1$

Step 5.2.3.1.3

Factor $4$ out of $4 (- x) + 4 (- 1)$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{5 (\frac{4 (- x - 1)}{5})}{4} - 1$

Step 5.2.3.2

Combine $5$ and $\frac{4 (- x - 1)}{5}$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{\frac{5 (4 (- x - 1))}{5}}{4} - 1$

Step 5.2.3.3

Multiply $5$ by $4$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{\frac{20 (- x - 1)}{5}}{4} - 1$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{20 (- x - 1)}{5}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $5$ out of $20 (- x - 1)$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{\frac{5 (4 (- x - 1))}{5}}{4} - 1$

Step 5.2.3.4.1.2

Factor $5$ out of $5$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{\frac{5 (4 (- x - 1))}{5 (1)}}{4} - 1$

Step 5.2.3.4.1.3

Cancel the common factor.

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{\frac{5 (4 (- x - 1))}{5 \cdot 1}}{4} - 1$

Step 5.2.3.4.1.4

Rewrite the expression.

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{\frac{4 (- x - 1)}{1}}{4} - 1$

Step 5.2.3.4.2

Divide $4 (- x - 1)$ by $1$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{4 (- x - 1)}{4} - 1$

Step 5.2.3.5

Cancel the common factor of $4$ .

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Step 5.2.3.5.1

Cancel the common factor.

$g^{- 1} (\frac{- 4 x - 4}{5}) = - \frac{4 (- x - 1)}{4} - 1$

Step 5.2.3.5.2

Divide $- x - 1$ by $1$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = - (- x - 1) - 1$

Step 5.2.3.6

Apply the distributive property.

$g^{- 1} (\frac{- 4 x - 4}{5}) = x + 1 - 1$

Step 5.2.3.7

Multiply $- - x$ .

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Step 5.2.3.7.1

Multiply $- 1$ by $- 1$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = 1 x + 1 - 1$

Step 5.2.3.7.2

Multiply $x$ by $1$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = x + 1 - 1$

Step 5.2.3.8

Multiply $- 1$ by $- 1$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = x + 1 - 1$

Step 5.2.4

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

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Step 5.2.4.1

Subtract $1$ from $1$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$g^{- 1} (\frac{- 4 x - 4}{5}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $g (- \frac{5 x}{4} - 1)$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (- \frac{5 x}{4} - 1) = \frac{- 4 (- \frac{5 x}{4} - 1) - 4}{5}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $- 4$ out of $- 4 (- \frac{5 x}{4} - 1) - 4$ .

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Step 5.3.3.1.1

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

$g (- \frac{5 x}{4} - 1) = \frac{- 4 (- \frac{5 x}{4} - 1) - 4 \cdot 1}{5}$

Step 5.3.3.1.2

Factor $- 4$ out of $- 4 (- \frac{5 x}{4} - 1) - 4 (1)$ .

$g (- \frac{5 x}{4} - 1) = \frac{- 4 (- \frac{5 x}{4} - 1 + 1)}{5}$

Step 5.3.3.2

Add $- 1$ and $1$ .

$g (- \frac{5 x}{4} - 1) = \frac{- 4 (- \frac{5 x}{4} + 0)}{5}$

Step 5.3.3.3

Add $- \frac{5 x}{4}$ and $0$ .

$g (- \frac{5 x}{4} - 1) = \frac{- 4 \cdot (- 1 \frac{5 x}{4})}{5}$

Step 5.3.3.4

Combine exponents.

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Step 5.3.3.4.1

Factor out negative.

$g (- \frac{5 x}{4} - 1) = \frac{- (- 4 \frac{5 x}{4})}{5}$

Step 5.3.3.4.2

Combine $- 4$ and $\frac{5 x}{4}$ .

$g (- \frac{5 x}{4} - 1) = \frac{- \frac{- 4 (5 x)}{4}}{5}$

Step 5.3.3.4.3

Multiply $5$ by $- 4$ .

$g (- \frac{5 x}{4} - 1) = \frac{- \frac{- 20 x}{4}}{5}$

Step 5.3.3.5

Cancel the common factor of $- 20$ and $4$ .

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Step 5.3.3.5.1

Factor $4$ out of $- 20 x$ .

$g (- \frac{5 x}{4} - 1) = \frac{- \frac{4 (- 5 x)}{4}}{5}$

Step 5.3.3.5.2

Cancel the common factors.

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Step 5.3.3.5.2.1

Factor $4$ out of $4$ .

$g (- \frac{5 x}{4} - 1) = \frac{- \frac{4 (- 5 x)}{4 (1)}}{5}$

Step 5.3.3.5.2.2

Cancel the common factor.

$g (- \frac{5 x}{4} - 1) = \frac{- \frac{4 (- 5 x)}{4 \cdot 1}}{5}$

Step 5.3.3.5.2.3

Rewrite the expression.

$g (- \frac{5 x}{4} - 1) = \frac{- \frac{- 5 x}{1}}{5}$

Step 5.3.3.5.2.4

Divide $- 5 x$ by $1$ .

$g (- \frac{5 x}{4} - 1) = \frac{- (- 5 x)}{5}$

$g (- \frac{5 x}{4} - 1) = \frac{5 x}{5}$

Step 5.3.3.6

Multiply $- 1$ by $- 5$ .

$g (- \frac{5 x}{4} - 1) = \frac{5 x}{5}$

Step 5.3.4

Cancel the common factor of $5$ .

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Step 5.3.4.1

Cancel the common factor.

$g (- \frac{5 x}{4} - 1) = \frac{5 x}{5}$

Step 5.3.4.2

Divide $x$ by $1$ .

$g (- \frac{5 x}{4} - 1) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = - \frac{5 x}{4} - 1$ is the inverse of $g (x) = \frac{- 4 x - 4}{5}$ .

$g^{- 1} (x) = - \frac{5 x}{4} - 1$