Precalculus Examples

$g (x) = \frac{- x - 5}{3}$

Step 1

Write $g (x) = \frac{- x - 5}{3}$ as an equation.

$y = \frac{- x - 5}{3}$

Step 2

Interchange the variables.

$x = \frac{- y - 5}{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{- y - 5}{3} = x$ .

$\frac{- y - 5}{3} = x$

Step 3.2

Multiply both sides by $3$ .

$\frac{- y - 5}{3} \cdot 3 = x \cdot 3$

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

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{- y - 5}{3} \cdot 3 = x \cdot 3$

Step 3.3.1.1.2

Rewrite the expression.

$- y - 5 = x \cdot 3$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

$- y - 5 = 3 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $5$ to both sides of the equation.

$- y = 3 x + 5$

Step 3.4.2

Divide each term in $- y = 3 x + 5$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- y = 3 x + 5$ by $- 1$ .

$\frac{- y}{- 1} = \frac{3 x}{- 1} + \frac{5}{- 1}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{3 x}{- 1} + \frac{5}{- 1}$

Step 3.4.2.2.2

Divide $y$ by $1$ .

$y = \frac{3 x}{- 1} + \frac{5}{- 1}$

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 one from the denominator of $\frac{3 x}{- 1}$ .

$y = - 1 \cdot (3 x) + \frac{5}{- 1}$

Step 3.4.2.3.1.2

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

$y = - (3 x) + \frac{5}{- 1}$

Step 3.4.2.3.1.3

Multiply $3$ by $- 1$ .

$y = - 3 x + \frac{5}{- 1}$

Step 3.4.2.3.1.4

Divide $5$ by $- 1$ .

$y = - 3 x - 5$

Step 4

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

$g^{- 1} (x) = - 3 x - 5$

Step 5

Verify if $g^{- 1} (x) = - 3 x - 5$ is the inverse of $g (x) = \frac{- x - 5}{3}$ .

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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 $g^{- 1} (\frac{- x - 5}{3})$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (\frac{- x - 5}{3}) = - 3 \frac{- x - 5}{3} - 5$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $3$ .

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Step 5.2.3.1.1

Factor $3$ out of $- 3$ .

$g^{- 1} (\frac{- x - 5}{3}) = 3 (- 1) (\frac{- x - 5}{3}) - 5$

Step 5.2.3.1.2

Cancel the common factor.

$g^{- 1} (\frac{- x - 5}{3}) = 3 \cdot (- 1 \frac{- x - 5}{3}) - 5$

Step 5.2.3.1.3

Rewrite the expression.

$g^{- 1} (\frac{- x - 5}{3}) = - 1 (- x - 5) - 5$

Step 5.2.3.2

Apply the distributive property.

$g^{- 1} (\frac{- x - 5}{3}) = - 1 (- x) - 1 \cdot - 5 - 5$

Step 5.2.3.3

Multiply $- 1 (- x)$ .

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Step 5.2.3.3.1

Multiply $- 1$ by $- 1$ .

$g^{- 1} (\frac{- x - 5}{3}) = 1 x - 1 \cdot - 5 - 5$

Step 5.2.3.3.2

Multiply $x$ by $1$ .

$g^{- 1} (\frac{- x - 5}{3}) = x - 1 \cdot - 5 - 5$

Step 5.2.3.4

Multiply $- 1$ by $- 5$ .

$g^{- 1} (\frac{- x - 5}{3}) = x + 5 - 5$

Step 5.2.4

Combine the opposite terms in $x + 5 - 5$ .

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Step 5.2.4.1

Subtract $5$ from $5$ .

$g^{- 1} (\frac{- x - 5}{3}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$g^{- 1} (\frac{- x - 5}{3}) = x$

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 $g (- 3 x - 5)$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (- 3 x - 5) = \frac{- (- 3 x - 5) - 5}{3}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $- 1$ out of $- (- 3 x - 5) - 5$ .

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Step 5.3.3.1.1

Rewrite $- 5$ as $- 1 (5)$ .

$g (- 3 x - 5) = \frac{- (- 3 x - 5) - 1 \cdot 5}{3}$

Step 5.3.3.1.2

Factor $- 1$ out of $- (- 3 x - 5) - 1 (5)$ .

$g (- 3 x - 5) = \frac{- (- 3 x - 5 + 5)}{3}$

Step 5.3.3.2

Add $- 5$ and $5$ .

$g (- 3 x - 5) = \frac{- (- 3 x + 0)}{3}$

Step 5.3.3.3

Add $- 3 x$ and $0$ .

$g (- 3 x - 5) = \frac{3 x}{3}$

Step 5.3.3.4

Multiply $- 1$ by $- 3$ .

$g (- 3 x - 5) = \frac{3 x}{3}$

Step 5.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.1

Cancel the common factor.

$g (- 3 x - 5) = \frac{3 x}{3}$

Step 5.3.4.2

Divide $x$ by $1$ .

$g (- 3 x - 5) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = - 3 x - 5$ is the inverse of $g (x) = \frac{- x - 5}{3}$ .

$g^{- 1} (x) = - 3 x - 5$